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+# Self-Supervised Learning with Kernel Dependence Maximization
+
+Yazhe Li∗ DeepMind and Gatsby Unit, UCL yazhe@google.com
+
+Roman Pogodin∗ Gatsby Unit, UCL roman.pogodin.17@ucl.ac.uk
+
+Danica J. Sutherland UBC and Amii† dsuth@cs.ubc.ca
+
+Arthur Gretton Gatsby Unit, UCL arthur.gretton@gmail.com
+
+# Abstract
+
+We approach self-supervised learning of image representations from a statistical dependence perspective, proposing Self-Supervised Learning with the HilbertSchmidt Independence Criterion (SSL-HSIC). SSL-HSIC maximizes dependence between representations of transformations of an image and the image identity, while minimizing the kernelized variance of those representations. This framework yields a new understanding of InfoNCE, a variational lower bound on the mutual information (MI) between different transformations. While the MI itself is known to have pathologies which can result in learning meaningless representations, its bound is much better behaved: we show that it implicitly approximates SSLHSIC (with a slightly different regularizer). Our approach also gives us insight into BYOL, a negative-free SSL method, since SSL-HSIC similarly learns local neighborhoods of samples. SSL-HSIC allows us to directly optimize statistical dependence in time linear in the batch size, without restrictive data assumptions or indirect mutual information estimators. Trained with or without a target network, SSL-HSIC matches the current state-of-the-art for standard linear evaluation on ImageNet [1], semi-supervised learning and transfer to other classification and vision tasks such as semantic segmentation, depth estimation and object recognition. Code is available at https://github.com/deepmind/ssl_hsic.
+
+# 1 Introduction
+
+Learning general-purpose visual representations without human supervision is a long-standing goal of machine learning. Specifically, we wish to find a feature extractor that captures the image semantics of a large unlabeled collection of images, so that e.g. various image understanding tasks can be achieved with simple linear models. One approach takes the latent representation of a likelihood-based generative model [2–8]; such models, though, solve a harder problem than necessary since semantic features need not capture low-level details of the input. Another option is to train a self-supervised model for a “pretext task,” such as predicting the position of image patches, identifying rotations, or image inpainting [9–14]. Designing good pretext tasks, however, is a subtle art, with little theoretical guidance available. Recently, a class of models based on contrastive learning [15–22] has seen substantial success: dataset images are cropped, rotated, color shifted, etc. into several views, and features are then trained to pull together representations of the “positive” pairs of views of the same source image, and push apart those of “negative” pairs (from different images). These methods are either understood from an information theoretic perspective as estimating the mutual information between the “positives” [15], or explained as aligning features subject to a uniformity constraint [23]. Another line of research [24, 25] attempts to learn representation without the “negative” pairs, but requires either a target network or stop-gradient operation to avoid collapsing.
+
+![](images/da54d49a0b8c54cb3816387abb70ecdac5d5f210aa0b1718b9848f10ffbf779d.jpg)  
+Figure 1: Top-1 accuracies with linear evaluation for different ResNet architecture and methods: supervised (as in [25]), SSL-HSIC (with a target network; ours), BYOL [25], SwAV [18], SimCLR [19], MoCo v2 [20] and Barlow Twins [22].
+
+![](images/9ef83fb128305bc0710f24b07be35a85254f2c88c20040144f68367e800a834c.jpg)  
+Figure 2: Statistical dependence view of contrastive learning: representations of transformed images should highly depend on image identity. Measuring dependence with HSIC, this pushes different images’ representation distributions apart (black arrows) and pulls representations of the same image together (colored shapes).
+
+We examine the contrastive framework from a statistical dependence point of view: feature representations for a given transformed image should be highly dependent on the image identity (Figure 2). To measure dependence, we turn to the Hilbert-Schmidt Independence Criterion (HSIC) [26], and propose a new loss for self-supervised learning which we call SSL-HSIC. Our loss is inspired by HSIC Bottleneck [27, 28], an alternative to Information Bottleneck [29], where we use the image identity as the label, but change the regularization term.
+
+Through the dependence maximization perspective, we present a unified view of various selfsupervised losses. Previous work [30] has shown that the success of InfoNCE cannot be solely attributed to properties of mutual information, in particular because mutual information (unlike kernel measures of dependence) has no notion of geometry in feature space: for instance, all invertible encoders achieve maximal mutual information, but they can output dramatically different representations with very different downstream performance [30]. Variational bounds on mutual information do impart notions of locality that allow them to succeed in practice, departing from the mutual information quantity that they try to estimate. We prove that InfoNCE, a popular such bound, in fact approximates SSL-HSIC with a variance-based regularization. Thus, InfoNCE can be thought of as working because it implicitly estimates a kernel-based notion of dependence. We additionally show SSL-HSIC is related to metric learning, where the features learn to align to the structure induced by the self-supervised labels. This perspective is closely related to the objective of BYOL [25], and can explain properties such as alignment and uniformity [23] observed in contrastive learning.
+
+Our perspective brings additional advantages, in computation and in simplicity of the algorithm, compared with existing approaches. Unlike the indirect variational bounds on mutual information [15, 31, 32], SSL-HSIC can be directly estimated from mini-batches of data. Unlike “negative-free” methods, the SSL-HSIC loss itself penalizes trivial solutions, so techniques such as target networks are not needed for reasonable outcomes. Using a target network does improve the performance of our method, however, suggesting target networks have other advantages that are not yet well understood. Finally, we employ random Fourier features [33] in our implementation, resulting in cost linear in batch size.
+
+Our main contributions are as follows:
+
+• We introduce SSL-HSIC, a principled self-supervised loss using kernel dependence maximization. • We present a unified view of contrastive learning through dependence maximization, by establishing relationships between SSL-HSIC, InfoNCE, and metric learning.
+
+• Our method achieves top-1 accuracy of $7 4 . 8 \%$ and top-5 accuracy of $9 2 . 2 \%$ with linear evaluations (see Figure 1 for a comparison with other methods), top-1 accuracy of $8 0 . 2 \%$ and Top-5 accuracy of $9 4 . 7 \%$ with fine-tuning, and competitive performance on a diverse set of downstream tasks.
+
+# 2 Background
+
+# 2.1 Self-supervised learning
+
+Recent developments in self-supervised learning, such as contrastive learning, try to ensure that features of two random views of an image are more associated with each other than with random views of other images. Typically, this is done through some variant of a classification loss, with one “positive” pair and many “negatives.” Other methods can learn solely from “positive” pairs, however. There have been many variations of this general framework in the past few years.
+
+Van den Oord et al. [15] first formulated the InfoNCE loss, which estimates a lower bound of the mutual information between the feature and the context. SimCLR [19, 34] carefully investigates the contribution of different data augmentations, and scales up the training batch size to include more negative examples. MoCo [17] increases the number of negative examples by using a memory bank. BYOL [25] learns solely on positive image pairs, training so that representations of one view match that of the other under a moving average of the featurizer. Instead of the moving average, SimSiam [24] suggests a stop-gradient on one of the encoders is enough to prevent BYOL from finding trivial solutions. SwAV [18] clusters the representation online, and uses distance from the cluster centers rather than computing pairwise distances of the data. Barlow Twins [22] uses an objective related to the cross-correlation matrix of the two views, motivated by redundancy reduction. It is perhaps the most related to our work in the literature (and their covariance matrix can be connected to HSIC [35]), but our method measures dependency more directly. While Barlow Twins decorrelates components of final representations, we maximize the dependence between the image’s abstract identity and its transformations.
+
+On the theory side, InfoNCE is proposed as a variational bound on Mutual Information between the representation of two views of the same image [15, 32]. Tschannen et al. [30] observe that InfoNCE performance cannot be explained solely by the properties of the mutual information, however, but is influenced more by other factors, such as the formulation of the estimator and the architecture of the feature extractor. Essentially, representations with the same MI can have drastically different representational qualities.
+
+To see this, consider a problem with two inputs, $A$ and $B$ (Figure 3, green and purple), and a one-dimensional featurizer, parameterized by the integer $M$ , which maps $A$ to Uniform $\langle \bar { \{ 0 , 2 , \ldots , 2 M \} } \rangle$ ) and $B$ to $\operatorname { U n i f o r m } ( \{ 1 , 3 , \dotsc , 2 M + 1 \} )$ . When $M \ = \ 0$ , the inputs are encoded into linearly separable features $A = 0$ and $B = 1$ (Figure 3, bottom). Otherwise when $M > 0$ , they are interspersed like $A B A B A B A B - { \mathfrak { a } }$ representation which is much harder to work with for downstream learners. Nevertheless, the mutual information between the features of any two augmentations of the same input (a positive pair) is independent of $M$ , that is $H [ Z _ { 1 } ] - H [ Z _ { 1 } \bar { | } Z _ { 2 } ] = \log 2$ for any $M$ . Note that InfoNCE would strongly prefer $M = 0$ , indeed behaving very differently from MI.
+
+![](images/ab4cec97781605937dcc3bb787c93d1942f1b0093dd330708f2ecba4418fd2a5.jpg)  
+Figure 3: Three distributions of positive examples for two classes (green and purple) that have the same mutual information, but drastically different quality for downstream learners.
+
+ater theories suggest that contrastive losses balance alignment of individual features and uniformity of the feature distribution [23], or in general alignment and some loss-defined distribution [36]. We propose to interpret the contrastive loss through the lens of statistical dependence, and relate it to metric learning, which naturally leads to alignment and uniformity.
+
+# 2.2 Hilbert-Schmidt Independence Criterion (HSIC)
+
+The Hilbert-Schmidt Independence Criterion (HSIC) [26] is a kernel-based measure of dependence between probability distributions. Like mutual information, for a wide range of kernels ${ \mathrm { H S I C } } ( X , Y ) = 0$ if and only if $X$ and $Y$ are independent [37], and large values of the measure correspond to “more dependence.” Unlike mutual information, HSIC incorporates a notion of geometry (via the kernel choice), and is both statistically and computationally easy to estimate. It has been used in a variety of applications, particularly for independence testing [38], but it has also been maximized in applications such as feature selection [39], clustering [40, 41], active learning [42], and as a classification loss called HSIC Bottleneck [27, 28] (similar ideas were expressed in [43, 44]).
+
+HSIC measures the dependence between two random variables by first taking a nonlinear feature transformation of each, say $\phi : \mathcal { X } \to \mathcal { F }$ and $\psi : \mathcal { V } \to \mathcal { G }$ (with $\mathcal { F }$ and $\mathcal { G }$ reproducing kernel Hilbert spaces, RKHSes1), and then evaluating the norm of the cross-covariance between those features:
+
+$$
+\operatorname { H S I C } ( X , Y ) = \left\| \mathbb { E } [ \phi ( X ) \psi ( Y ) ^ { \top } ] - \mathbb { E } [ \phi ( X ) ] \mathbb { E } [ \psi ( Y ) ] ^ { \top } \right\| _ { H S } ^ { 2 } .
+$$
+
+Here $\| \cdot \| _ { H S }$ is the Hilbert-Schmidt norm, which in finite dimensions is the usual Frobenius norm. HSIC measures the scale of the correlation in these nonlinear features, which allows it to identify nonlinear dependencies between $X$ and $Y$ with appropriate features $\phi$ and $\psi$ .
+
+Inner products in an RKHS are by definition kernel functions: $k ( x , x ^ { \prime } ) \ = \ \langle \phi ( x ) , \phi ( x ^ { \prime } ) \rangle _ { \mathcal { F } }$ and $l ( y , y ^ { \prime } ) = \langle \psi ( y ) , \psi ( y ^ { \prime } ) \rangle _ { \mathcal { G } }$ . Let $( X ^ { \prime } , Y ^ { \prime } )$ , $( X ^ { \prime \prime } , Y ^ { \prime \prime } )$ be independent copies of $( X , Y )$ ; this gives
+
+$$
+\operatorname { H S I C } ( X , Y ) = \operatorname { \mathbb { E } } \left[ k ( X , X ^ { \prime } ) l ( Y , Y ^ { \prime } ) \right] - 2 \operatorname { \mathbb { E } } \left[ k ( X , X ^ { \prime } ) l ( Y , Y ^ { \prime \prime } ) \right] + \operatorname { \mathbb { E } } \left[ k ( X , X ^ { \prime } ) \right] \operatorname { \mathbb { E } } \left[ l ( Y , Y ^ { \prime } ) \right] .
+$$
+
+HSIC is also straightforward to estimate: given i.i.d. samples $\left\{ ( x _ { 1 } , y _ { 1 } ) , \dotsc , ( x _ { N } , y _ { N } ) \right\}$ drawn i.i.d. from the joint distribution of $( X , Y )$ , Gretton et al. [26] propose an estimator
+
+$$
+{ \widehat { \mathrm { H S I C } } } ( X , Y ) = \frac { 1 } { ( N - 1 ) ^ { 2 } } \mathrm { T r } ( K H L H ) ,
+$$
+
+where $K _ { i j } = k ( x _ { i } , x _ { j } )$ and $L _ { i j } = l ( y _ { i } , y _ { j } )$ are the kernel matrices, and $\begin{array} { r } { H = I - \frac { 1 } { N } \mathbf { 1 } \mathbf { 1 } ^ { \top } } \end{array}$ is called the centering matrix. This estimator has an $\bar { O } ( 1 / N )$ bias, which is not a concern for our uses; however, an unbiased estimator with the same computational cost is available [39].
+
+# 3 Self-supervised learning with Kernel Dependence Maximization
+
+Our method builds on the self-supervised learning framework used by most of the recent selfsupervised learning approaches [16–19, 22, 24, 25]. For a dataset with $N$ points $x _ { i }$ , each point goes through a random transformation $t ^ { p } ( x _ { i } )$ (e.g. random crop), and then forms a feature representation $z _ { i } ^ { p } = { \bar { f } } _ { \theta } ( t ^ { p } ( x _ { i } ) )$ with an encoder network $f _ { \theta }$ . We associate each image $x _ { i }$ with its identity $y _ { i }$ , which works as a one-hot encoded label: $y _ { i } \in \mathbb { R } ^ { N }$ and $( y _ { i } ) _ { d } = 1$ iff $d = i$ (and zero otherwise). To match the transformations and image identities, we maximize the dependence between $z _ { i }$ and $y _ { i }$ such that $z _ { i }$ is predictive of its original image. To build representations suitable for downstream tasks, we also need to penalize high-variance representations. These ideas come together in our HSIC-based objective for self-supervised learning, which we term SSL-HSIC:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { S S L - H S I C } } ( \boldsymbol { \theta } ) = - \mathrm { H S I C } \left( \boldsymbol { Z } , \boldsymbol { Y } \right) + \gamma \sqrt { \mathrm { H S I C } \left( \boldsymbol { Z } , \boldsymbol { Z } \right) } . } \end{array}
+$$
+
+Unlike contrastive losses, which make the $z _ { i } ^ { p }$ from the same $x _ { i }$ closer and those from different $x _ { j }$ more distant, we propose an alternative way to match different transformations of the same image with its abstract identity (e.g. position in the dataset).
+
+Our objective also resembles the HSIC bottleneck for supervised learning [27] (in particular, the version of [28]), but ours uses a square root for $\mathrm { H S I C } ( Z , Z )$ . The square root makes the two terms on the same scale: $\mathrm { H S I C } ( Z , Y )$ is effectively a dot product, and $\sqrt { \mathrm { H S I C } ( Z , Z ) }$ a norm, so that e.g. scaling the kernel by a constant does not change the relative amount of regularization;2 this also gives better performance in practice.
+
+Due to the one-hot encoded labels, we can re-write HSIC $( Z , Y )$ as (see Appendix A)
+
+$$
+\begin{array} { r } { \mathrm { H S I C } ( Z , Y ) \propto \mathbb { E } _ { z _ { 1 } , z _ { 2 } \sim p o s } \left[ k ( z _ { 1 } , z _ { 2 } ) \right] - \mathbb { E } _ { z _ { 1 } } \mathbb { E } _ { z _ { 2 } } \left[ k ( z _ { 1 } , z _ { 2 } ) \right] , } \end{array}
+$$
+
+where the first expectation is over the distribution of “positive” pairs (those from the same source image), and the second one is a sum over all image pairs, including their transformations. The first term in (5) pushes representations belonging to the same image identity together, while the second term keeps mean representations for each identity apart (as in Figure 2). The scaling of $\mathrm { H S I C } ( Z , Y )$ depends on the choice of the kernel over $Y$ , and is irrelevant to the optimization.
+
+This form also reveals three key theoretical results. Section 3.1 shows that InfoNCE is better understood as an HSIC-based loss than a mutual information between views. Section 3.2 reveals that the dependence maximization in $\mathrm { H S I C } ( Z , Y )$ can also be viewed as a form of distance metric learning, where the cluster structure is defined by the labels. Finally, $\mathrm { H S I C } ( Z , Y )$ is proportional to the average kernel-based distance between the distribution of views for each source image (the maximum mean discrepancy, MMD; see Appendix B.2).
+
+# 3.1 Connection to InfoNCE
+
+In this section we show the connection between InfoNCE and our loss; see Appendix B for full details. We first write the latter in its infinite sample size limit (see [23] for a derivation) as
+
+$$
+\mathcal { L } _ { \mathrm { I n f o N C E } } ( \theta ) = - \mathbb { E } _ { z _ { 1 } , z _ { 2 } \sim \mathrm { p o s } } \left[ k ( z _ { 1 } , z _ { 2 } ) \right] + \mathbb { E } _ { z _ { 1 } } \log \mathbb { E } _ { z _ { 2 } } \left[ \exp \left( k ( z _ { 1 } , z _ { 2 } ) \right) \right] ,
+$$
+
+where the last two expectations are taken over all points, and the first is over the distribution of positive pairs. The kernel $k ( z _ { 1 } , z _ { 2 } )$ was originally formulated as a scoring function in a form of a dot product [15], and then a scaled cosine similarity [19]. Both functions are valid kernels.
+
+Now assume that $k ( z _ { 1 } , z _ { 2 } )$ doesn’t deviate much from $\mathbb { E } _ { z _ { 2 } } \left[ k ( z _ { 1 } , z _ { 2 } ) \right]$ , Taylor-expand the exponent in (6) around $\mathbb { E } _ { z _ { 2 } } \left[ k ( z _ { 1 } , z _ { 2 } ) \right]$ , then expand $\log ( 1 + \mathbb { E } _ { z _ { 2 } } ( . . . ) ) \approx \mathbb { E } _ { z _ { 2 } } ( . . . )$ . We obtain an $\mathrm { H S I C } ( Z , Y )$ - based objective:
+
+$$
+\mathcal { L } _ { \mathrm { I n f o N C E } } ( \theta ) \approx \underbrace { - \mathbb { E } _ { z _ { 1 } , z _ { 2 } \sim \mathrm { p o s } } \left[ k ( z _ { 1 } , z _ { 2 } ) \right] + \mathbb { E } _ { z _ { 1 } } \mathbb { E } _ { z _ { 2 } } \left[ k ( z _ { 1 } , z _ { 2 } ) \right] } _ { \propto - \mathrm { H S I C } ( Z , Y ) } + \underbrace { \frac { 1 } { 2 } \mathbb { E } _ { z _ { 1 } } \left[ \mathbb { V } \mathrm { a r } _ { z _ { 2 } } \left[ k ( z _ { 1 } , z _ { 2 } ) \right] \right] } _ { \mathrm { v a r i a n c e p e n a l t y } } .
+$$
+
+Since the scaling of $\mathrm { H S I C } ( Z , Y )$ is irrelevant to the optimization, we assume scaling to replace $\propto$ with $=$ . In the small variance regime, we can show that for the right $\gamma$ ,
+
+$$
+\begin{array} { r } { - \mathrm { H S I C } \left( Z , Y \right) + \gamma \mathrm { H S I C } \left( Z , Z \right) \le \mathcal { L } _ { \mathrm { I n f o N C E } } ( \theta ) + o ( \mathrm { v a r i a n c e } ) . } \end{array}
+$$
+
+For $\mathrm { H S I C } ( Z , Z ) \le 1$ , we also have that
+
+$$
+- \mathrm { H S I C } \left( Z , Y \right) + \gamma \mathrm { H S I C } \left( Z , Z \right) \leq \mathcal { L } _ { \mathrm { S S L - H S I C } } ( \theta )
+$$
+
+due to the square root. InfoNCE and SSL-HSIC in general don’t quite bound each other due to discrepancy in the variance terms, but in practice the difference is small.
+
+Why should we prefer the HSIC interpretation of InfoNCE? Initially, InfoNCE was suggested as a variational approximation to the mutual information between two views [15]. It has been observed, however, that using tighter estimators of mutual information leads to worse performance [30]. It is also simple to construct examples where InfoNCE finds different representations while the underlying MI remains constant [30]. Alternative theories suggest that InfoNCE balances alignment of “positive” examples and uniformity of the overall feature representation [23], or that (under strong assumptions) it can identify the latent structure in a hypothesized data-generating process, akin to nonlinear ICA [50]. Our view is consistent with these theories, but doesn’t put restrictive assumptions on the input data or learned representations. In Section 5 (summarized in Table 8a), we show that our interpretation gives rise to a better objective in practice.
+
+# 3.2 Connection to metric learning
+
+Our SSL-HSIC objective is closely related to kernel alignment [48], especially centered kernel alignment [49]. As a kernel method for distance metric learning, kernel alignment measures the agreement between a kernel function and a target function. Intuitively, the self-supervised labels $Y$ imply a cluster structure, and $\mathrm { H S I C } ( Z , Y )$ estimates the degree of agreement between the learned features and this cluster structure in the kernel space. This relationship with clustering is also established in [40, 41, 45], where labels are learned rather than features. The clustering perspective is more evident when we assume linear kernels over both $Z$ and $Y$ , and $Z$ is unit length and centered:3
+
+$$
+\begin{array} { r l } & { - \mathrm { H S I C } ( Z , Y ) \propto - \displaystyle \frac { 1 } { M } T r ( Y ^ { \top } Z ^ { \top } Z Y ) + T r ( Z ^ { \top } Z ) - N M } \\ & { \qquad = - \displaystyle \frac { 1 } { M } \sum _ { i = 1 } ^ { N } \left\| \sum _ { p = 1 } ^ { M } z _ { i } ^ { p } \right\| _ { 2 } ^ { 2 } + \sum _ { i = 1 } ^ { N } \sum _ { p = 1 } ^ { M } \| z _ { i } ^ { p } \| _ { 2 } ^ { 2 } - N M = \sum _ { i = 1 } ^ { N } \sum _ { p = 1 } ^ { M } \| z _ { i } ^ { p } - \bar { z } _ { i } \| _ { 2 } ^ { 2 } - N M , } \end{array}
+$$
+
+with $M$ the number of augmentations per image and $\bar { z } _ { i } = \sum _ { p } z _ { i } ^ { p } / M$ the average feature vector of the augmented views of $x _ { i }$ . We emphasize, though, that (10) assumes centered, normalized data with linear kernels; the right-hand side of (10) could be optimized by setting all $z _ { i } ^ { p }$ to the same vector for each $i$ , but this does not actually optimize $\mathrm { H S I C } ( Z , \bar { Y } )$ .
+
+Equation (10) shows that we recover the spectral formulation [51] and sum-of-squares loss used in the $\mathbf { k }$ -means clustering algorithm from the kernel objective. Moreover, the self-supervised label imposes that the features from transformations of the same image are gathered in the same cluster. Equation (10) also allows us to connect SSL-HSIC to non-contrastive objectives such as BYOL, although the connection is subtle because of its use of predictor and target networks. If each image is augmented with two views, we can compute (10) using $\bar { z _ { i } } \approx ( z _ { i } ^ { 1 } + \bar { z _ { i } ^ { 2 } } ) / 2$ , so the clustering loss becomes $\propto \textstyle \sum _ { i } | | z _ { i } ^ { 1 } - z _ { i } ^ { 2 } | | _ { 2 } ^ { 2 }$ . This is exactly the BYOL objective, only that $z _ { i } ^ { 2 }$ in BYOL comes from a target network. The assumption of centered and normalized features for (10) is important in the case of BYOL: without it, BYOL can find trivial solutions where all the features are collapsed to the same feature vector far away from the origin. The target network is used to prevent the collapse. SSL-HSIC, on the other hand, rules out such a solution by building the centering into the loss function, and therefore can be trained successfully without a target network or stop gradient operation.
+
+# 3.3 Estimator of SSL-HSIC
+
+To use SSL-HSIC, we need to correctly and efficiently estimate (4). Both points are non-trivial: the self-supervised framework implies non-i.i.d. batches (due to positive examples), while the estimator in (3) assumes i.i.d. data; moreover, the time to compute (3) is quadratic in the batch size.
+
+First, for $\mathrm { H S I C } ( Z , Z )$ we use the biased estimator in (3). Although the i.i.d. estimator (3) results in an $O ( 1 / B )$ bias for $B$ original images in the batch size (see Appendix A), the batch size $B$ is large in our case and therefore the bias is negligible. For $\mathrm { H S I C } ( Z , Y )$ the situation is more delicate: the i.i.d. estimator needs re-scaling, and its bias depends on the number of positive examples $M$ , which is typically very small (usually 2). We propose the following estimator:
+
+$$
+\widehat { \mathrm { H S I C } } ( Z , Y ) = \frac { \Delta l } { N } \left( \frac { 1 } { B M ( M - 1 ) } \sum _ { i p l } k ( z _ { i } ^ { p } , z _ { i } ^ { l } ) - \frac { 1 } { B ^ { 2 } M ^ { 2 } } \sum _ { i j p l } k ( z _ { i } ^ { p } , z _ { j } ^ { l } ) - \frac { 1 } { M - 1 } \right) ,
+$$
+
+where $i$ and $j$ index original images, and $p$ and $l$ their random transformations; $k$ is the kernel used for latent $Z , l$ is the kernel used for the labels, and $\Delta l = l ( i , i ) - l ( i , j )$ ( $\it l$ for same labels minus $l$ for different labels). Note that due to the one-hot structure of self-supervised labels $Y$ , the standard (i.i.d.-based) estimator would miss the $1 / N$ scaling and the $M - 1$ correction (the latter is important in practice, as we usually have $M = 2$ ). See Appendix A for the derivations.
+
+For convenience, we assume $\Delta l = N$ (any scaling of $l$ can be subsumed by $\gamma$ ), and optimize
+
+$$
+\widehat { \mathcal { L } } _ { \mathrm { S S L - H S I C } } ( \theta ) = - \widehat { \mathrm { H S I C } } ( Z , Y ) + \gamma \sqrt { \widehat { \mathrm { H S I C } } ( Z , Z ) } .
+$$
+
+The computational complexity of the proposed estimators is $O ( B ^ { 2 } M ^ { 2 } )$ for each mini-batch of size $B$ with $M$ augmentations. We can reduce the complexity to $O ( B M )$ by using random Fourier features (RFF) [33], which approximate the kernel $k ( z _ { 1 } , z _ { 2 } )$ with a carefully chosen random $D$ -dimensional approximation $R ( z _ { 1 } ) ^ { \top } R ( z _ { 2 } )$ for $R ( z ) : \mathbb { R } ^ { D _ { z } }  \mathbb { R } ^ { D }$ , such that $\begin{array} { r } { k ( z _ { 1 } , z _ { 2 } ) = \mathbb { E } \left[ R ( z _ { 1 } ) ^ { \top } R ( z _ { 2 } ) \right] , } \end{array}$ . Fourier frequencies are sampled independently for the two kernels on $Z$ in $\mathrm { H S I C } ( Z , Z )$ at each training step. We leave the details on how to construct $R ( z )$ for the kernels we use to Appendix C.
+
+# 4 Experiments
+
+In this section, we present our experimental setup, where we assess the performance of the representation learned with SSL-HSIC both with and without a target network. First, we train a model with a standard ResNet-50 backbone using SSL-HSIC as objective on the training set of ImageNet ILSVRC-2012 [1]. For evaluation, we retain the backbone as a feature extractor for downstream tasks. We evaluate the representation on various downstream tasks including classification, object segmentation, object detection and depth estimation.
+
+![](images/a05cd99484a48004d410b54e0a447b170aabd0f27381d90ac49f4fcc73067ee3.jpg)  
+Figure 4: Architecture and SSL-HSIC objective. A self-supervised label $y$ – an indicator of the image identity – is associated with an image $x$ . Image transformation functions $t$ are sampled and applied to the original image, resulting in views $t ^ { 1 } ( x )$ and $t ^ { 2 } ( x )$ . Features $z ^ { 1 }$ and $z ^ { 2 }$ are obtained after passing the augmented views through encoder $( f )$ , projector $( g )$ , and possibly predictor $( q )$ networks, while label $y$ is retained. Kernel matrices, $K$ for the latents and $L$ for the labels, are computed on the mini-batch of data; SSL-HSIC is estimated with $K$ and $L$ as in (12). The blue boxes reflect two potential options: when using a target network, $\xi$ is a moving average of $\theta$ , and a predictor network $q$ is added; without the target network, $q$ is removed and $\xi$ is simply equal to $\theta$ .
+
+# 4.1 Implementation
+
+Architecture Figure 4 illustrates the architecture we used for SSL-HSIC in this section. To facilitate comparison between different methods, our encoder $f _ { \theta }$ uses the standard ResNet-50 backbone without the final classification layer. The output of the encoder is a 2048-dimension embedding vector, which is the representation used for downstream tasks. As in BYOL [25], our projector $g$ and predictor $q$ networks are 2-layer MLPs with 4096 hidden dimensions and 256 output dimensions. The outputs of the networks are batch-normalized and rescaled to unit norm before computing the loss. We use an inverse multiquadric kernel (IMQ) for the latent representation (approximated with 512 random Fourier features that are resampled at each step; see Appendix C for details) and a linear kernel for labels. $\gamma$ in (4) is set to 3. When training without a target network, unlike SimSiam [24], we do not stop gradients for either branch. If the target network is used, its weights are an exponential moving average of the online network weights. We employ the same schedule as BYOL [25], $\tau = 1 \stackrel { - } { - } 0 . 0 1 \stackrel { - } { \cdot } ( \cos ( \pi t / T ) + 1 ) / 2$ with $t$ the current step and $T$ the total training steps.
+
+Image augmentation Our method uses the same data augmentation scheme as BYOL (see Appendix D.1). Briefly, we first draw a random patch from the original image and resize it to $2 2 4 \times 2 2 4$ Then, we apply a random horizontal flip, followed by color jittering, consisting of a random sequence of brightness, contrast, saturation, hue adjustments, and an optional grayscale conversion. Finally Gaussian blur and solarization are applied, and the view is normalized with ImageNet statistics.
+
+Optimization We train the model with a batch size of 4096 on 128 Cloud TPU v4 cores. Again, following [19, 25], we use the LARS optimizer [52] with a cosine decay learning rate schedule over 1000 epochs. The base learning rate to all of our experiments is 0.4 and it is scaled linearly [53] with the batch size $l r = 0 . 4 \times b a t c h { \_ } s i z e / 2 5 6 .$ . All experiments use weight decay of $1 0 ^ { - 6 }$ .
+
+Learning kernel parameters We use a linear kernel for labels, since the type of kernel only scales (12). Our inverse multiquadric kernel for the latent $Z$ has an additional kernel scale parameter. We optimize this along with all other parameters, but regularize it to maximize the entropy of the distribution $k _ { \sigma } ( s )$ , where $s _ { i j } = \lVert { z } _ { i } - { z } _ { j } \rVert ^ { 2 }$ ; this amounts to maximizing $\log \lVert k _ { \sigma } ^ { \prime } ( s ) \rVert ^ { 2 }$ (Appendix D.1.2).
+
+Linear evaluation on ImageNet Learned features are evaluated with the standard linear evaluation protocol commonly used in evaluating self-supervised learning methods [15–22, 25]. Table 1 reports the top-1 and top-5 accuracies obtained with SSL-HSIC on ImageNet validation set, and compares to previous self-supervised learning methods. Without a target network, our method reaches $7 2 . 2 \%$ top-1 and $9 0 . 7 \%$ top-5 accuracies. Unlike BYOL, the SSL-HSIC objective prevents the network from finding trivial solutions as explained in Section 3.2. Adding the target network, our method outperforms most previous methods, achieving top-1 accuracy of $7 4 . 8 \%$ and top-5 accuracy of $9 2 . 2 \%$ The fact that we see performance gains from adopting a target network suggests that its effect is not yet well understood, although note discussion in [25] which points to its stabilizing effect.
+
+Table 1: Linear evaluation on the ImageNet validation set.   
+
+<table><tr><td></td><td>Top-1(%)</td><td>Top-5(%)</td></tr><tr><td>Supervised [54]</td><td>75.9</td><td>92.8</td></tr><tr><td>SimCLR[19]</td><td>69.3</td><td>89.0</td></tr><tr><td>MoCo v2 [20]</td><td>71.1</td><td>90.1</td></tr><tr><td>BYOL [25]</td><td>74.3</td><td>91.6</td></tr><tr><td>SwAV[18]</td><td>75.3</td><td>-</td></tr><tr><td>Barlow Twins [22]</td><td>73.2</td><td>91.0</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>72.2</td><td>90.7</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>74.8</td><td>92.2</td></tr></table>
+
+Table 2: Fine-tuning on $1 \%$ , $10 \%$ and $100 \%$ of the ImageNet training set and evaluating on the validation set.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Top-1(%)</td><td colspan="3">Top-5(%)</td></tr><tr><td>1%</td><td>10%</td><td>100%</td><td>1%</td><td>10%</td><td>100%</td></tr><tr><td>Supervised [54]</td><td>25.4</td><td>56.4</td><td>75.9</td><td>48.4</td><td>80.4</td><td>92.8</td></tr><tr><td>SimCLR [19]</td><td>48.3</td><td>65.6</td><td>76.0</td><td>75.5</td><td>87.8</td><td>93.1</td></tr><tr><td>BYOL [25]</td><td>53.2</td><td>68.8</td><td>77.7</td><td>78.4</td><td>89.0</td><td>93.9</td></tr><tr><td>SwAV[18]</td><td>53.9</td><td>70.2</td><td>1</td><td>78.5</td><td>89.9</td><td>-</td></tr><tr><td>Barlow Twins [22]</td><td>55.0</td><td>69.7</td><td>-</td><td>79.2</td><td>89.3</td><td>-</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>45.3</td><td>65.5</td><td>76.4</td><td>72.7</td><td>87.5</td><td>93.2</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>52.1</td><td>67.9</td><td>77.2</td><td>77.7</td><td>88.6</td><td>93.6</td></tr></table>
+
+Semi-supervised learning on ImageNet We fine-tune the network pretrained with SSL-HSIC on $1 \%$ , $10 \%$ and $100 \%$ of ImageNet, using the same ImageNet splits as SimCLR [19]. Table 2 summarizes the semi-supervised learning performance. Our method, with or without a target network, has competitive performance in both data regimes. The target network has the most impact on the small-data regime, with $1 \%$ labels.
+
+Table 3: Comparison of transfer learning performance on 12 image datasets. Supervised-IN is trained on ImageNet with supervised pretrainining. Random init trains on individual dataset with randomly initialized weights. MPCA refers to mean per-class accuracy; AP50 is average precision at $\mathrm { I o U } { = } 0 . 5$ .   
+
+<table><tr><td>Dataset Metric</td><td>Birdsnap Caltechl01 Cifarl0 Cifarl001 Top-1</td><td>MPCA</td><td></td><td>Top-1</td><td>DTD Top-1</td><td>Aircraft Food Flowers</td><td>MPCA Top-1</td><td>MPCA MPCA Top-1</td><td>Pets</td><td>Cars</td><td>Top-1</td><td>SUN397 VOC2007 AP50</td></tr><tr><td>Linear:</td><td></td><td></td><td>Top-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Supervised-IN [19]</td><td>53.7</td><td>94.5</td><td>93.6</td><td>78.3</td><td></td><td>61.0</td><td>72.3</td><td>94.7</td><td>91.5</td><td>67.8</td><td>61.9</td><td>82.8</td></tr><tr><td>SimCLR[19]</td><td>37.4</td><td>90.3</td><td>90.6</td><td>71.6</td><td>74.9 74.5</td><td>50.3</td><td>68.4</td><td>90.3</td><td>83.6</td><td>50.3</td><td>58.8</td><td>80.5</td></tr><tr><td>BYOL [25]</td><td>57.2</td><td>94.2</td><td>91.3</td><td>78.4</td><td>75.5</td><td>60.6</td><td>75.3</td><td>96.1</td><td>90.4</td><td>66.7</td><td>62.2</td><td>82.5</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>50.6</td><td>92.3</td><td>91.5</td><td>75.9</td><td>75.3</td><td>57.9</td><td>73.6</td><td>95.0</td><td>88.2</td><td>59.3</td><td>61.0</td><td>81.4</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>57.8</td><td>93.5</td><td>92.3</td><td>77.0</td><td>76.2</td><td>58.5</td><td>75.6</td><td>95.4</td><td>91.2</td><td>62.6</td><td>61.8</td><td>83.3</td></tr><tr><td>Fine-tuned:</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><tr><td>Supervised-IN [19]</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>85.0</td></tr><tr><td>Random init [19]</td><td>75.8</td><td>93.3</td><td>97.5</td><td>86.4</td><td>74.6</td><td>86.0 85.9</td><td>88.3</td><td>97.6</td><td>92.1</td><td>92.1</td><td>94.3 53.6</td><td>67.3</td></tr><tr><td>SimCLR[19]</td><td>76.1 75.9</td><td>72.6 92.1</td><td>95.9 97.7</td><td>80.2 85.9</td><td>64.8 73.2</td><td>88.1</td><td>86.9 88.2</td><td>92.0 97.0</td><td>81.5 89.2</td><td>91.4 91.3</td><td>63.5</td><td>84.1</td></tr><tr><td>BYOL [25]</td><td>76.3</td><td>93.8</td><td>97.8</td><td>86.1</td><td>76.2</td><td>88.1</td><td>88.5</td><td>97.0</td><td>91.7</td><td>91.6</td><td>63.7</td><td>85.4</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>73.1</td><td>91.5</td><td>97.4</td><td>85.3</td><td>75.3</td><td>87.1</td><td>87.5</td><td>96.4</td><td>90.6</td><td>91.6</td><td>62.2</td><td>84.1</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>74.9</td><td>93.8</td><td>97.8</td><td>84.7</td><td>75.4</td><td>88.9</td><td>87.7</td><td>97.3</td><td>91.7</td><td>91.8</td><td>61.7</td><td>84.1</td></tr></table>
+
+Transfer to other classification tasks To investigate the generality of the representation learned with SSL-HSIC, we evaluate the transfer performance for classification on 12 natural image datasets [55–64] using the same procedure as [19, 25, 65]. Table 3 shows the top-1 accuracy of the linear evaluation and fine-tuning performance on the test set. SSL-HSIC gets state-of-the-art performance on 3 of the classification tasks and reaches strong performance on others for this benchmark, indicating the learned representations are robust for transfer learning.
+
+Transfer to other vision tasks To test the ability of transferring to tasks other than classification, we fine-tune the network on semantic segmentation, depth estimation and object detection tasks. We use Pascal VOC2012 dataset [58] for semantic segmentation, NYU v2 dataset [66] for depth estimation and COCO [67] for object detection. Object detection outputs either bounding box or object segmentation (instance segmentation). Details of the evaluations setup is in Appendix D.2. Table 4 and Table 5 shows that SSL-HSIC achieves competitive performance on all three vision tasks.
+
+Table 4: Fine-tuning performance on semantic segmentation and depth estimation. Mean Intersection over Union (mIoU) is reported for semantic segmentation. Relative error (rel), root mean squared error (rms), and the percent of pixels (pct) where the error is below $1 . 2 5 ^ { n }$ thresholds are reported for depth estimation.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">VOC2012 mIoU</td><td colspan="5">NYU v2</td></tr><tr><td>pct.&lt;1.25</td><td>pct.&lt;1.252</td><td>pct.&lt;1.253</td><td>rms</td><td>rel</td></tr><tr><td>Supervised-IN</td><td>74.4</td><td>81.1</td><td>95.3</td><td>98.8</td><td>0.573</td><td>0.127</td></tr><tr><td>SimCLR</td><td>75.2</td><td>83.3</td><td>96.5</td><td>99.1</td><td>0.557</td><td>0.134</td></tr><tr><td>BYOL</td><td>76.3</td><td>84.6</td><td>96.7</td><td>99.1</td><td>0.541</td><td>0.129</td></tr><tr><td>SSL-HSIC(w/o target)</td><td>74.9</td><td>84.1</td><td>96.7</td><td>99.2</td><td>0.539</td><td>0.130</td></tr><tr><td>SSL-HSIC(w/ target)</td><td>76.0</td><td>83.8</td><td>96.8</td><td>99.1</td><td>0.548</td><td>0.130</td></tr></table>
+
+Table 5: Fine-tuning performance on COCO object detection tasks. Precision, averaged over $1 0 \ \mathrm { I o U }$ (Intersection over Union) thresholds, is reported for both bounding box and object segmentation.   
+
+<table><tr><td>Method</td><td>APbb</td><td>Apmk</td></tr><tr><td>Supervised</td><td>39.6</td><td>35.6</td></tr><tr><td>SimCLR</td><td>39.7</td><td>35.8</td></tr><tr><td>MoCo v2</td><td>40.1</td><td>36.3</td></tr><tr><td>BYOL</td><td>41.6</td><td>37.2</td></tr><tr><td>SwAV</td><td>41.6</td><td>37.8</td></tr><tr><td>SSL-HSIC(w/o target)</td><td>40.5</td><td>36.3</td></tr><tr><td>SSL-HSIC(w/ target)</td><td>41.3</td><td>36.8</td></tr></table>
+
+# 5 Ablation Studies
+
+We present ablation studies to gain more intuition on SSL-HSIC. Here, we use a ResNet-50 backbone trained for 100 epochs on ImageNet, and evaluate with the linear protocol unless specified.
+
+ResNet architectures In this ablation, we investigate the performance of SSL-HSIC with wider and deeper ResNet architecture. Figure 1 and Table 6 show our main results. The performance of SSL-HSIC gets better with larger networks. We used the supervised baseline from [25] which our training framework is based on ([19] reports lower performance). The performance gap between SSL-HSIC and the supervised baseline diminishes with larger architectures. In addition, Table 7 presents the semi-supervise learning results with subsets $1 \%$ , $10 \%$ and $100 \%$ of the ImageNet data.
+
+Table 6: Top-1 and top-5 accuracies for different ResNet architectures using linear evaluation protocol.   
+
+<table><tr><td></td><td colspan="2">SSL-HSIC</td><td colspan="2">BYOL[25]</td><td colspan="2">Sup.[25]</td></tr><tr><td>ResNet</td><td>Top1</td><td>Top5</td><td>Top1</td><td>Top5</td><td>Top1</td><td>Top5</td></tr><tr><td>50 (1x)</td><td>74.8</td><td>92.2</td><td>74.3</td><td>91.6</td><td>76.4</td><td>92.9</td></tr><tr><td>50 (2x)</td><td>77.9</td><td>94.0</td><td>77.4</td><td>93.6</td><td>79.9</td><td>95.0</td></tr><tr><td>50(4x)</td><td>79.1</td><td>94.5</td><td>78.6</td><td>94.2</td><td>80.7</td><td>95.3</td></tr><tr><td>200 (2x)</td><td>79.6</td><td>94.8</td><td>79.6</td><td>94.9</td><td>80.1</td><td>95.2</td></tr></table>
+
+Table 7: Top-1 and top-5 accuracies for different ResNet architectures using semisupervised fine-tuning.   
+
+<table><tr><td></td><td colspan="3">Top1</td><td colspan="3">Top5</td></tr><tr><td>ResNet</td><td>1%</td><td>10%</td><td>100%</td><td>1%</td><td>10%</td><td>100%</td></tr><tr><td>50 (1x)</td><td>52.1</td><td>67.9</td><td>77.2</td><td>77.7</td><td>88.6</td><td>93.6</td></tr><tr><td>50 (2x)</td><td>61.2</td><td>72.6</td><td>79.3</td><td>83.8</td><td>91.2</td><td>94.7</td></tr><tr><td>50 (4x)</td><td>67.0</td><td>75.4</td><td>79.7</td><td>87.4</td><td>92.5</td><td>94.8</td></tr><tr><td>200(2x)</td><td>69.0</td><td>76.3</td><td>80.5</td><td>88.3</td><td>92.9</td><td>95.2</td></tr></table>
+
+Regularization term We compare performance of InfoNCE with SSL-HSIC in Table 8a since they can be seen as approximating the same $\mathrm { H S I C } ( Z , Y )$ objective but with different forms of regularization. We reproduce the InfoNCE result in our codebase, using the same architecture and data augmentiation as for SSL-HSIC. Trained for 100 epochs (without a target network), InfoNCE achieves $6 6 . 0 \%$ top-1 and $8 6 . 9 \%$ top-5 accuracies, which is better than the result reported in [19]. For comparison, SSL-HSIC reaches $6 6 . 7 \%$ top-1 and $8 7 . 6 \%$ top-5 accuracies. This suggests that the regularization employed by SSL-HSIC is more effective.
+
+Kernel type We investigate the effect of using different a kernel on latents $Z$ . Training without a target network or random Fourier feature approximation, the top-1 accuracies for linear, Gaussian, and inverse multiquadric (IMQ) kernels are $6 5 . 2 7 \%$ , $6 6 . 6 7 \%$ and $6 6 . 7 2 \%$ respectively. Non-linear kernels indeed improve the performance; Gaussian and IMQ kernels reach very similar performance for 100 epochs. We choose IMQ kernel for longer runs, because its heavy-tail property can capture more signal when points are far apart.
+
+Number of RFF Features Table 8b shows the performance of SSL-HSIC with different numbers of Fourier features. The RFF approximation has a minor impact on the overall performance, as long as we resample them; fixed sets of features performed poorly. Our main result picked 512 features, for substantial computational savings with minor loss in accuracy.
+
+Batch size Similar to most of the self-supervised learning methods [19, 25], SSL-HSIC benefits from using a larger batch size during training. However, the drop of performance from using smaller batch size is not as pronounced as it is in SimCLR[19] as shown in Table 8c.
+
+(a) Regularization   
+
+<table><tr><td></td><td>Top-1</td><td>Top-5</td></tr><tr><td>SSL-HSIC</td><td>66.7</td><td>87.6</td></tr><tr><td>InfoNCE</td><td>66.0</td><td>86.9</td></tr></table>
+
+Table 8: Linear evaluation results when varying different hyperparameters.   
+(d) Projector/predictor size   
+
+<table><tr><td>Output Dim</td><td>Top-1(%)</td></tr><tr><td>64</td><td>65.4</td></tr><tr><td>128</td><td>66.0</td></tr><tr><td>256</td><td>66.4</td></tr><tr><td>512</td><td>66.6</td></tr><tr><td>1024</td><td>66.6</td></tr></table>
+
+(b) # Fourier features   
+
+<table><tr><td>#RFFs</td><td>Top-1(%)</td></tr><tr><td>64</td><td>66.0</td></tr><tr><td>128</td><td>66.2</td></tr><tr><td>256</td><td>66.2</td></tr><tr><td>512</td><td>66.4</td></tr><tr><td>1024</td><td>66.5</td></tr><tr><td>2048</td><td>66.5</td></tr><tr><td>No Approx.</td><td>66.7</td></tr></table>
+
+(c) Batch size   
+
+<table><tr><td colspan="4">Top-1(%)</td></tr><tr><td>Batch Size</td><td>SSL-HSIC</td><td>SimCLR</td><td></td></tr><tr><td>256</td><td>63.7</td><td>57.5</td><td></td></tr><tr><td>512</td><td>65.6</td><td></td><td>60.7</td></tr><tr><td>1024</td><td>66.7</td><td></td><td>62.8</td></tr><tr><td>2048</td><td>67.1</td><td></td><td>64.0</td></tr><tr><td>4096</td><td>66.7</td><td></td><td>64.6</td></tr></table>
+
+Projector and predictor output size Table 8d shows the performance when using different output dimension for the projector/predictor networks. The performance saturates at 512 dimensions.
+
+# 6 Conclusions
+
+We introduced SSL-HSIC, a loss function for self-supervised representation learning based on kernel dependence maximization. We provided a unified view on various self-supervised learning losses: we proved that InfoNCE, a lower bound of mutual information, actually approximates SSL-HSIC with a variance-based regularization, and we can also interpret SSL-HSIC as metric learning where the cluster structure is imposed by the self-supervised label, of which the BYOL objective is a special case. We showed that training with SSL-HSIC achieves performance on par with the state-of-the-art on the standard self-supervised benchmarks.
+
+Although using the image identity as self-supervised label provides a good inductive bias, it might not be wholly satisfactory; we expect that some images pairs are in fact more similar than others, based e.g. on their ImageNet class label. It will be interesting to explore methods that combine label structure discovery with representation learning (as in SwAV [18]). In this paper, we only explored learning image representations, but in future work SSL-HSIC can be extended to learning structure for $Y$ as well, building on existing work [41, 45].
+
+# Broader impact
+
+Our work concentrates on providing a more theoretically grounded and interpretable loss function for self-supervised learning. A better understanding of self-supervised learning, especially through more interpretable learning dynamics, is likely to lead to better and more explicit control over societal biases of these algorithms. SSL-HSIC yields an alternative, clearer understanding of existing selfsupervised methods. As such, it is unlikely that our method introduces further biases than those already present for self-supervised learning.
+
+The broader impacts of the self-supervised learning framework is an area that has not been studied by the AI ethics community, but we think it calls for closer inspection. An important concern for fairness of ML algorithms is dataset bias. ImageNet is known for a number of problems such as offensive annotations, non-visual concepts and lack of diversity, in particular for underrepresented groups. Existing works and remedies typically focus on label bias. Since SSL doesn’t use labels, however, the type and degree of bias could be very different from that of supervised learning. To mitigate the risk of dataset bias, one could employ dataset re-balancing to correct sampling bias [68] or completely exclude human images from the dataset while achieving the same performance [69].
+
+A new topic to investigate for self-supervised learning is how the bias/unbiased representation could be transferred to downstream tasks. We are not aware of any work in this direction. Another area of concern is security and robustness. Compared to supervised learning, self-supervised learning typically involves more intensive data augmentation such as color jittering, brightness adjustment, etc. There is some initial evidence suggesting self-supervised learning improves model robustness [70]. However, since data augmentation can either be beneficial [71] or detrimental [72] depending on the type of adversarial attacks, more studies are needed to assess its role for self-supervised learning.
+
+# Acknowledgments and Disclosure of Funding
+
+The authors would like to thank Olivier J. Hénaff for valuable feedback on the manuscript and the help for evaluating object detection task. We thank Aaron Van den Oord and Oriol Vinyals for providing valuable feedback on the manuscript. We are grateful to Yonglong Tian, Ting Chen and the BYOL authors for the help with reproducing baselines and evaluating downstream tasks.
+
+This work was supported by DeepMind, the Gatsby Charitable Foundation, the Wellcome Trust, NSERC, and the Canada CIFAR AI Chairs program.
+
+# References
+
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parse/train/0HW7A5YZjq7/0HW7A5YZjq7_content_list.json

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+    {
+        "type": "text",
+        "text": "Self-Supervised Learning with Kernel Dependence Maximization ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Yazhe Li∗ DeepMind and Gatsby Unit, UCL yazhe@google.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Roman Pogodin∗ Gatsby Unit, UCL roman.pogodin.17@ucl.ac.uk ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Danica J. Sutherland UBC and Amii† dsuth@cs.ubc.ca ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Arthur Gretton Gatsby Unit, UCL arthur.gretton@gmail.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We approach self-supervised learning of image representations from a statistical dependence perspective, proposing Self-Supervised Learning with the HilbertSchmidt Independence Criterion (SSL-HSIC). SSL-HSIC maximizes dependence between representations of transformations of an image and the image identity, while minimizing the kernelized variance of those representations. This framework yields a new understanding of InfoNCE, a variational lower bound on the mutual information (MI) between different transformations. While the MI itself is known to have pathologies which can result in learning meaningless representations, its bound is much better behaved: we show that it implicitly approximates SSLHSIC (with a slightly different regularizer). Our approach also gives us insight into BYOL, a negative-free SSL method, since SSL-HSIC similarly learns local neighborhoods of samples. SSL-HSIC allows us to directly optimize statistical dependence in time linear in the batch size, without restrictive data assumptions or indirect mutual information estimators. Trained with or without a target network, SSL-HSIC matches the current state-of-the-art for standard linear evaluation on ImageNet [1], semi-supervised learning and transfer to other classification and vision tasks such as semantic segmentation, depth estimation and object recognition. Code is available at https://github.com/deepmind/ssl_hsic. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning general-purpose visual representations without human supervision is a long-standing goal of machine learning. Specifically, we wish to find a feature extractor that captures the image semantics of a large unlabeled collection of images, so that e.g. various image understanding tasks can be achieved with simple linear models. One approach takes the latent representation of a likelihood-based generative model [2–8]; such models, though, solve a harder problem than necessary since semantic features need not capture low-level details of the input. Another option is to train a self-supervised model for a “pretext task,” such as predicting the position of image patches, identifying rotations, or image inpainting [9–14]. Designing good pretext tasks, however, is a subtle art, with little theoretical guidance available. Recently, a class of models based on contrastive learning [15–22] has seen substantial success: dataset images are cropped, rotated, color shifted, etc. into several views, and features are then trained to pull together representations of the “positive” pairs of views of the same source image, and push apart those of “negative” pairs (from different images). These methods are either understood from an information theoretic perspective as estimating the mutual information between the “positives” [15], or explained as aligning features subject to a uniformity constraint [23]. Another line of research [24, 25] attempts to learn representation without the “negative” pairs, but requires either a target network or stop-gradient operation to avoid collapsing. ",
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+        "type": "image",
+        "img_path": "images/da54d49a0b8c54cb3816387abb70ecdac5d5f210aa0b1718b9848f10ffbf779d.jpg",
+        "image_caption": [
+            "Figure 1: Top-1 accuracies with linear evaluation for different ResNet architecture and methods: supervised (as in [25]), SSL-HSIC (with a target network; ours), BYOL [25], SwAV [18], SimCLR [19], MoCo v2 [20] and Barlow Twins [22]. "
+        ],
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+        "img_path": "images/9ef83fb128305bc0710f24b07be35a85254f2c88c20040144f68367e800a834c.jpg",
+        "image_caption": [
+            "Figure 2: Statistical dependence view of contrastive learning: representations of transformed images should highly depend on image identity. Measuring dependence with HSIC, this pushes different images’ representation distributions apart (black arrows) and pulls representations of the same image together (colored shapes). "
+        ],
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+    {
+        "type": "text",
+        "text": "We examine the contrastive framework from a statistical dependence point of view: feature representations for a given transformed image should be highly dependent on the image identity (Figure 2). To measure dependence, we turn to the Hilbert-Schmidt Independence Criterion (HSIC) [26], and propose a new loss for self-supervised learning which we call SSL-HSIC. Our loss is inspired by HSIC Bottleneck [27, 28], an alternative to Information Bottleneck [29], where we use the image identity as the label, but change the regularization term. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Through the dependence maximization perspective, we present a unified view of various selfsupervised losses. Previous work [30] has shown that the success of InfoNCE cannot be solely attributed to properties of mutual information, in particular because mutual information (unlike kernel measures of dependence) has no notion of geometry in feature space: for instance, all invertible encoders achieve maximal mutual information, but they can output dramatically different representations with very different downstream performance [30]. Variational bounds on mutual information do impart notions of locality that allow them to succeed in practice, departing from the mutual information quantity that they try to estimate. We prove that InfoNCE, a popular such bound, in fact approximates SSL-HSIC with a variance-based regularization. Thus, InfoNCE can be thought of as working because it implicitly estimates a kernel-based notion of dependence. We additionally show SSL-HSIC is related to metric learning, where the features learn to align to the structure induced by the self-supervised labels. This perspective is closely related to the objective of BYOL [25], and can explain properties such as alignment and uniformity [23] observed in contrastive learning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our perspective brings additional advantages, in computation and in simplicity of the algorithm, compared with existing approaches. Unlike the indirect variational bounds on mutual information [15, 31, 32], SSL-HSIC can be directly estimated from mini-batches of data. Unlike “negative-free” methods, the SSL-HSIC loss itself penalizes trivial solutions, so techniques such as target networks are not needed for reasonable outcomes. Using a target network does improve the performance of our method, however, suggesting target networks have other advantages that are not yet well understood. Finally, we employ random Fourier features [33] in our implementation, resulting in cost linear in batch size. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contributions are as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We introduce SSL-HSIC, a principled self-supervised loss using kernel dependence maximization. • We present a unified view of contrastive learning through dependence maximization, by establishing relationships between SSL-HSIC, InfoNCE, and metric learning. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Our method achieves top-1 accuracy of $7 4 . 8 \\%$ and top-5 accuracy of $9 2 . 2 \\%$ with linear evaluations (see Figure 1 for a comparison with other methods), top-1 accuracy of $8 0 . 2 \\%$ and Top-5 accuracy of $9 4 . 7 \\%$ with fine-tuning, and competitive performance on a diverse set of downstream tasks. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 Background ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "2.1 Self-supervised learning ",
+        "text_level": 1,
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Recent developments in self-supervised learning, such as contrastive learning, try to ensure that features of two random views of an image are more associated with each other than with random views of other images. Typically, this is done through some variant of a classification loss, with one “positive” pair and many “negatives.” Other methods can learn solely from “positive” pairs, however. There have been many variations of this general framework in the past few years. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Van den Oord et al. [15] first formulated the InfoNCE loss, which estimates a lower bound of the mutual information between the feature and the context. SimCLR [19, 34] carefully investigates the contribution of different data augmentations, and scales up the training batch size to include more negative examples. MoCo [17] increases the number of negative examples by using a memory bank. BYOL [25] learns solely on positive image pairs, training so that representations of one view match that of the other under a moving average of the featurizer. Instead of the moving average, SimSiam [24] suggests a stop-gradient on one of the encoders is enough to prevent BYOL from finding trivial solutions. SwAV [18] clusters the representation online, and uses distance from the cluster centers rather than computing pairwise distances of the data. Barlow Twins [22] uses an objective related to the cross-correlation matrix of the two views, motivated by redundancy reduction. It is perhaps the most related to our work in the literature (and their covariance matrix can be connected to HSIC [35]), but our method measures dependency more directly. While Barlow Twins decorrelates components of final representations, we maximize the dependence between the image’s abstract identity and its transformations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On the theory side, InfoNCE is proposed as a variational bound on Mutual Information between the representation of two views of the same image [15, 32]. Tschannen et al. [30] observe that InfoNCE performance cannot be explained solely by the properties of the mutual information, however, but is influenced more by other factors, such as the formulation of the estimator and the architecture of the feature extractor. Essentially, representations with the same MI can have drastically different representational qualities. ",
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+    {
+        "type": "text",
+        "text": "To see this, consider a problem with two inputs, $A$ and $B$ (Figure 3, green and purple), and a one-dimensional featurizer, parameterized by the integer $M$ , which maps $A$ to Uniform $\\langle \\bar { \\{ 0 , 2 , \\ldots , 2 M \\} } \\rangle$ ) and $B$ to $\\operatorname { U n i f o r m } ( \\{ 1 , 3 , \\dotsc , 2 M + 1 \\} )$ . When $M \\ = \\ 0$ , the inputs are encoded into linearly separable features $A = 0$ and $B = 1$ (Figure 3, bottom). Otherwise when $M > 0$ , they are interspersed like $A B A B A B A B - { \\mathfrak { a } }$ representation which is much harder to work with for downstream learners. Nevertheless, the mutual information between the features of any two augmentations of the same input (a positive pair) is independent of $M$ , that is $H [ Z _ { 1 } ] - H [ Z _ { 1 } \\bar { | } Z _ { 2 } ] = \\log 2$ for any $M$ . Note that InfoNCE would strongly prefer $M = 0$ , indeed behaving very differently from MI. ",
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+    {
+        "type": "image",
+        "img_path": "images/ab4cec97781605937dcc3bb787c93d1942f1b0093dd330708f2ecba4418fd2a5.jpg",
+        "image_caption": [
+            "Figure 3: Three distributions of positive examples for two classes (green and purple) that have the same mutual information, but drastically different quality for downstream learners. "
+        ],
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+    {
+        "type": "text",
+        "text": "ater theories suggest that contrastive losses balance alignment of individual features and uniformity of the feature distribution [23], or in general alignment and some loss-defined distribution [36]. We propose to interpret the contrastive loss through the lens of statistical dependence, and relate it to metric learning, which naturally leads to alignment and uniformity. ",
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+    {
+        "type": "text",
+        "text": "2.2 Hilbert-Schmidt Independence Criterion (HSIC) ",
+        "text_level": 1,
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+        "text": "The Hilbert-Schmidt Independence Criterion (HSIC) [26] is a kernel-based measure of dependence between probability distributions. Like mutual information, for a wide range of kernels ${ \\mathrm { H S I C } } ( X , Y ) = 0$ if and only if $X$ and $Y$ are independent [37], and large values of the measure correspond to “more dependence.” Unlike mutual information, HSIC incorporates a notion of geometry (via the kernel choice), and is both statistically and computationally easy to estimate. It has been used in a variety of applications, particularly for independence testing [38], but it has also been maximized in applications such as feature selection [39], clustering [40, 41], active learning [42], and as a classification loss called HSIC Bottleneck [27, 28] (similar ideas were expressed in [43, 44]). ",
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+        "type": "text",
+        "text": "HSIC measures the dependence between two random variables by first taking a nonlinear feature transformation of each, say $\\phi : \\mathcal { X } \\to \\mathcal { F }$ and $\\psi : \\mathcal { V } \\to \\mathcal { G }$ (with $\\mathcal { F }$ and $\\mathcal { G }$ reproducing kernel Hilbert spaces, RKHSes1), and then evaluating the norm of the cross-covariance between those features: ",
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+        "img_path": "images/852eeb6eec8aea02dd762c218f0b47878f503bd29f222cac5531791bdcad1a4e.jpg",
+        "text": "$$\n\\operatorname { H S I C } ( X , Y ) = \\left\\| \\mathbb { E } [ \\phi ( X ) \\psi ( Y ) ^ { \\top } ] - \\mathbb { E } [ \\phi ( X ) ] \\mathbb { E } [ \\psi ( Y ) ] ^ { \\top } \\right\\| _ { H S } ^ { 2 } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here $\\| \\cdot \\| _ { H S }$ is the Hilbert-Schmidt norm, which in finite dimensions is the usual Frobenius norm. HSIC measures the scale of the correlation in these nonlinear features, which allows it to identify nonlinear dependencies between $X$ and $Y$ with appropriate features $\\phi$ and $\\psi$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "Inner products in an RKHS are by definition kernel functions: $k ( x , x ^ { \\prime } ) \\ = \\ \\langle \\phi ( x ) , \\phi ( x ^ { \\prime } ) \\rangle _ { \\mathcal { F } }$ and $l ( y , y ^ { \\prime } ) = \\langle \\psi ( y ) , \\psi ( y ^ { \\prime } ) \\rangle _ { \\mathcal { G } }$ . Let $( X ^ { \\prime } , Y ^ { \\prime } )$ , $( X ^ { \\prime \\prime } , Y ^ { \\prime \\prime } )$ be independent copies of $( X , Y )$ ; this gives ",
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+        "type": "equation",
+        "img_path": "images/6668a57c492fe39c9144befae6affc7590fe099e16e51f228cd2ac5532cc40bb.jpg",
+        "text": "$$\n\\operatorname { H S I C } ( X , Y ) = \\operatorname { \\mathbb { E } } \\left[ k ( X , X ^ { \\prime } ) l ( Y , Y ^ { \\prime } ) \\right] - 2 \\operatorname { \\mathbb { E } } \\left[ k ( X , X ^ { \\prime } ) l ( Y , Y ^ { \\prime \\prime } ) \\right] + \\operatorname { \\mathbb { E } } \\left[ k ( X , X ^ { \\prime } ) \\right] \\operatorname { \\mathbb { E } } \\left[ l ( Y , Y ^ { \\prime } ) \\right] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "HSIC is also straightforward to estimate: given i.i.d. samples $\\left\\{ ( x _ { 1 } , y _ { 1 } ) , \\dotsc , ( x _ { N } , y _ { N } ) \\right\\}$ drawn i.i.d. from the joint distribution of $( X , Y )$ , Gretton et al. [26] propose an estimator ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/c8ba4024f5e40415b8d989ceeda424434c1d5582a842624dd05fb4ed154afaa3.jpg",
+        "text": "$$\n{ \\widehat { \\mathrm { H S I C } } } ( X , Y ) = \\frac { 1 } { ( N - 1 ) ^ { 2 } } \\mathrm { T r } ( K H L H ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $K _ { i j } = k ( x _ { i } , x _ { j } )$ and $L _ { i j } = l ( y _ { i } , y _ { j } )$ are the kernel matrices, and $\\begin{array} { r } { H = I - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ { \\top } } \\end{array}$ is called the centering matrix. This estimator has an $\\bar { O } ( 1 / N )$ bias, which is not a concern for our uses; however, an unbiased estimator with the same computational cost is available [39]. ",
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+        "type": "text",
+        "text": "3 Self-supervised learning with Kernel Dependence Maximization ",
+        "text_level": 1,
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+        "text": "Our method builds on the self-supervised learning framework used by most of the recent selfsupervised learning approaches [16–19, 22, 24, 25]. For a dataset with $N$ points $x _ { i }$ , each point goes through a random transformation $t ^ { p } ( x _ { i } )$ (e.g. random crop), and then forms a feature representation $z _ { i } ^ { p } = { \\bar { f } } _ { \\theta } ( t ^ { p } ( x _ { i } ) )$ with an encoder network $f _ { \\theta }$ . We associate each image $x _ { i }$ with its identity $y _ { i }$ , which works as a one-hot encoded label: $y _ { i } \\in \\mathbb { R } ^ { N }$ and $( y _ { i } ) _ { d } = 1$ iff $d = i$ (and zero otherwise). To match the transformations and image identities, we maximize the dependence between $z _ { i }$ and $y _ { i }$ such that $z _ { i }$ is predictive of its original image. To build representations suitable for downstream tasks, we also need to penalize high-variance representations. These ideas come together in our HSIC-based objective for self-supervised learning, which we term SSL-HSIC: ",
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+    {
+        "type": "equation",
+        "img_path": "images/377b837f625ac02e81309b037a786373d99855b97c66b98accd179e1ff903615.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { S S L - H S I C } } ( \\boldsymbol { \\theta } ) = - \\mathrm { H S I C } \\left( \\boldsymbol { Z } , \\boldsymbol { Y } \\right) + \\gamma \\sqrt { \\mathrm { H S I C } \\left( \\boldsymbol { Z } , \\boldsymbol { Z } \\right) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Unlike contrastive losses, which make the $z _ { i } ^ { p }$ from the same $x _ { i }$ closer and those from different $x _ { j }$ more distant, we propose an alternative way to match different transformations of the same image with its abstract identity (e.g. position in the dataset). ",
+        "bbox": [
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+        "type": "text",
+        "text": "Our objective also resembles the HSIC bottleneck for supervised learning [27] (in particular, the version of [28]), but ours uses a square root for $\\mathrm { H S I C } ( Z , Z )$ . The square root makes the two terms on the same scale: $\\mathrm { H S I C } ( Z , Y )$ is effectively a dot product, and $\\sqrt { \\mathrm { H S I C } ( Z , Z ) }$ a norm, so that e.g. scaling the kernel by a constant does not change the relative amount of regularization;2 this also gives better performance in practice. ",
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+    {
+        "type": "text",
+        "text": "Due to the one-hot encoded labels, we can re-write HSIC $( Z , Y )$ as (see Appendix A) ",
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+    {
+        "type": "equation",
+        "img_path": "images/0208221dffe7712d774ffaf6c530bb7b763087d9478695aa452ece0cf6179e2e.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { H S I C } ( Z , Y ) \\propto \\mathbb { E } _ { z _ { 1 } , z _ { 2 } \\sim p o s } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right] - \\mathbb { E } _ { z _ { 1 } } \\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the first expectation is over the distribution of “positive” pairs (those from the same source image), and the second one is a sum over all image pairs, including their transformations. The first term in (5) pushes representations belonging to the same image identity together, while the second term keeps mean representations for each identity apart (as in Figure 2). The scaling of $\\mathrm { H S I C } ( Z , Y )$ depends on the choice of the kernel over $Y$ , and is irrelevant to the optimization. ",
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+        "type": "text",
+        "text": "This form also reveals three key theoretical results. Section 3.1 shows that InfoNCE is better understood as an HSIC-based loss than a mutual information between views. Section 3.2 reveals that the dependence maximization in $\\mathrm { H S I C } ( Z , Y )$ can also be viewed as a form of distance metric learning, where the cluster structure is defined by the labels. Finally, $\\mathrm { H S I C } ( Z , Y )$ is proportional to the average kernel-based distance between the distribution of views for each source image (the maximum mean discrepancy, MMD; see Appendix B.2). ",
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+    {
+        "type": "text",
+        "text": "3.1 Connection to InfoNCE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we show the connection between InfoNCE and our loss; see Appendix B for full details. We first write the latter in its infinite sample size limit (see [23] for a derivation) as ",
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+        "type": "equation",
+        "img_path": "images/3f0f16bba114be5d8c5c03e090df921e66943d3f2aee43cefc166db7c1941606.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { I n f o N C E } } ( \\theta ) = - \\mathbb { E } _ { z _ { 1 } , z _ { 2 } \\sim \\mathrm { p o s } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right] + \\mathbb { E } _ { z _ { 1 } } \\log \\mathbb { E } _ { z _ { 2 } } \\left[ \\exp \\left( k ( z _ { 1 } , z _ { 2 } ) \\right) \\right] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the last two expectations are taken over all points, and the first is over the distribution of positive pairs. The kernel $k ( z _ { 1 } , z _ { 2 } )$ was originally formulated as a scoring function in a form of a dot product [15], and then a scaled cosine similarity [19]. Both functions are valid kernels. ",
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+        "type": "text",
+        "text": "Now assume that $k ( z _ { 1 } , z _ { 2 } )$ doesn’t deviate much from $\\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right]$ , Taylor-expand the exponent in (6) around $\\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right]$ , then expand $\\log ( 1 + \\mathbb { E } _ { z _ { 2 } } ( . . . ) ) \\approx \\mathbb { E } _ { z _ { 2 } } ( . . . )$ . We obtain an $\\mathrm { H S I C } ( Z , Y )$ - based objective: ",
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+        "text": "$$\n\\mathcal { L } _ { \\mathrm { I n f o N C E } } ( \\theta ) \\approx \\underbrace { - \\mathbb { E } _ { z _ { 1 } , z _ { 2 } \\sim \\mathrm { p o s } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right] + \\mathbb { E } _ { z _ { 1 } } \\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right] } _ { \\propto - \\mathrm { H S I C } ( Z , Y ) } + \\underbrace { \\frac { 1 } { 2 } \\mathbb { E } _ { z _ { 1 } } \\left[ \\mathbb { V } \\mathrm { a r } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right] \\right] } _ { \\mathrm { v a r i a n c e p e n a l t y } } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Since the scaling of $\\mathrm { H S I C } ( Z , Y )$ is irrelevant to the optimization, we assume scaling to replace $\\propto$ with $=$ . In the small variance regime, we can show that for the right $\\gamma$ , ",
+        "bbox": [
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+        "text": "$$\n\\begin{array} { r } { - \\mathrm { H S I C } \\left( Z , Y \\right) + \\gamma \\mathrm { H S I C } \\left( Z , Z \\right) \\le \\mathcal { L } _ { \\mathrm { I n f o N C E } } ( \\theta ) + o ( \\mathrm { v a r i a n c e } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For $\\mathrm { H S I C } ( Z , Z ) \\le 1$ , we also have that ",
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+    {
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+        "text": "$$\n- \\mathrm { H S I C } \\left( Z , Y \\right) + \\gamma \\mathrm { H S I C } \\left( Z , Z \\right) \\leq \\mathcal { L } _ { \\mathrm { S S L - H S I C } } ( \\theta )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "due to the square root. InfoNCE and SSL-HSIC in general don’t quite bound each other due to discrepancy in the variance terms, but in practice the difference is small. ",
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+        "text": "Why should we prefer the HSIC interpretation of InfoNCE? Initially, InfoNCE was suggested as a variational approximation to the mutual information between two views [15]. It has been observed, however, that using tighter estimators of mutual information leads to worse performance [30]. It is also simple to construct examples where InfoNCE finds different representations while the underlying MI remains constant [30]. Alternative theories suggest that InfoNCE balances alignment of “positive” examples and uniformity of the overall feature representation [23], or that (under strong assumptions) it can identify the latent structure in a hypothesized data-generating process, akin to nonlinear ICA [50]. Our view is consistent with these theories, but doesn’t put restrictive assumptions on the input data or learned representations. In Section 5 (summarized in Table 8a), we show that our interpretation gives rise to a better objective in practice. ",
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+        "type": "text",
+        "text": "3.2 Connection to metric learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our SSL-HSIC objective is closely related to kernel alignment [48], especially centered kernel alignment [49]. As a kernel method for distance metric learning, kernel alignment measures the agreement between a kernel function and a target function. Intuitively, the self-supervised labels $Y$ imply a cluster structure, and $\\mathrm { H S I C } ( Z , Y )$ estimates the degree of agreement between the learned features and this cluster structure in the kernel space. This relationship with clustering is also established in [40, 41, 45], where labels are learned rather than features. The clustering perspective is more evident when we assume linear kernels over both $Z$ and $Y$ , and $Z$ is unit length and centered:3 ",
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+        "text": "$$\n\\begin{array} { r l } & { - \\mathrm { H S I C } ( Z , Y ) \\propto - \\displaystyle \\frac { 1 } { M } T r ( Y ^ { \\top } Z ^ { \\top } Z Y ) + T r ( Z ^ { \\top } Z ) - N M } \\\\ & { \\qquad = - \\displaystyle \\frac { 1 } { M } \\sum _ { i = 1 } ^ { N } \\left\\| \\sum _ { p = 1 } ^ { M } z _ { i } ^ { p } \\right\\| _ { 2 } ^ { 2 } + \\sum _ { i = 1 } ^ { N } \\sum _ { p = 1 } ^ { M } \\| z _ { i } ^ { p } \\| _ { 2 } ^ { 2 } - N M = \\sum _ { i = 1 } ^ { N } \\sum _ { p = 1 } ^ { M } \\| z _ { i } ^ { p } - \\bar { z } _ { i } \\| _ { 2 } ^ { 2 } - N M , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "with $M$ the number of augmentations per image and $\\bar { z } _ { i } = \\sum _ { p } z _ { i } ^ { p } / M$ the average feature vector of the augmented views of $x _ { i }$ . We emphasize, though, that (10) assumes centered, normalized data with linear kernels; the right-hand side of (10) could be optimized by setting all $z _ { i } ^ { p }$ to the same vector for each $i$ , but this does not actually optimize $\\mathrm { H S I C } ( Z , \\bar { Y } )$ . ",
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+    {
+        "type": "text",
+        "text": "Equation (10) shows that we recover the spectral formulation [51] and sum-of-squares loss used in the $\\mathbf { k }$ -means clustering algorithm from the kernel objective. Moreover, the self-supervised label imposes that the features from transformations of the same image are gathered in the same cluster. Equation (10) also allows us to connect SSL-HSIC to non-contrastive objectives such as BYOL, although the connection is subtle because of its use of predictor and target networks. If each image is augmented with two views, we can compute (10) using $\\bar { z _ { i } } \\approx ( z _ { i } ^ { 1 } + \\bar { z _ { i } ^ { 2 } } ) / 2$ , so the clustering loss becomes $\\propto \\textstyle \\sum _ { i } | | z _ { i } ^ { 1 } - z _ { i } ^ { 2 } | | _ { 2 } ^ { 2 }$ . This is exactly the BYOL objective, only that $z _ { i } ^ { 2 }$ in BYOL comes from a target network. The assumption of centered and normalized features for (10) is important in the case of BYOL: without it, BYOL can find trivial solutions where all the features are collapsed to the same feature vector far away from the origin. The target network is used to prevent the collapse. SSL-HSIC, on the other hand, rules out such a solution by building the centering into the loss function, and therefore can be trained successfully without a target network or stop gradient operation. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 Estimator of SSL-HSIC ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To use SSL-HSIC, we need to correctly and efficiently estimate (4). Both points are non-trivial: the self-supervised framework implies non-i.i.d. batches (due to positive examples), while the estimator in (3) assumes i.i.d. data; moreover, the time to compute (3) is quadratic in the batch size. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "First, for $\\mathrm { H S I C } ( Z , Z )$ we use the biased estimator in (3). Although the i.i.d. estimator (3) results in an $O ( 1 / B )$ bias for $B$ original images in the batch size (see Appendix A), the batch size $B$ is large in our case and therefore the bias is negligible. For $\\mathrm { H S I C } ( Z , Y )$ the situation is more delicate: the i.i.d. estimator needs re-scaling, and its bias depends on the number of positive examples $M$ , which is typically very small (usually 2). We propose the following estimator: ",
+        "bbox": [
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+            523,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e8584c88a53b312adc015e59cabde66c5f77f99650cb96ea0d2496add414a696.jpg",
+        "text": "$$\n\\widehat { \\mathrm { H S I C } } ( Z , Y ) = \\frac { \\Delta l } { N } \\left( \\frac { 1 } { B M ( M - 1 ) } \\sum _ { i p l } k ( z _ { i } ^ { p } , z _ { i } ^ { l } ) - \\frac { 1 } { B ^ { 2 } M ^ { 2 } } \\sum _ { i j p l } k ( z _ { i } ^ { p } , z _ { j } ^ { l } ) - \\frac { 1 } { M - 1 } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            597,
+            776,
+            646
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $i$ and $j$ index original images, and $p$ and $l$ their random transformations; $k$ is the kernel used for latent $Z , l$ is the kernel used for the labels, and $\\Delta l = l ( i , i ) - l ( i , j )$ ( $\\it l$ for same labels minus $l$ for different labels). Note that due to the one-hot structure of self-supervised labels $Y$ , the standard (i.i.d.-based) estimator would miss the $1 / N$ scaling and the $M - 1$ correction (the latter is important in practice, as we usually have $M = 2$ ). See Appendix A for the derivations. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For convenience, we assume $\\Delta l = N$ (any scaling of $l$ can be subsumed by $\\gamma$ ), and optimize ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b66475b9da98befa9943a5613b26dbb795453ced10e1f5b98427f81b5627b47d.jpg",
+        "text": "$$\n\\widehat { \\mathcal { L } } _ { \\mathrm { S S L - H S I C } } ( \\theta ) = - \\widehat { \\mathrm { H S I C } } ( Z , Y ) + \\gamma \\sqrt { \\widehat { \\mathrm { H S I C } } ( Z , Z ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The computational complexity of the proposed estimators is $O ( B ^ { 2 } M ^ { 2 } )$ for each mini-batch of size $B$ with $M$ augmentations. We can reduce the complexity to $O ( B M )$ by using random Fourier features (RFF) [33], which approximate the kernel $k ( z _ { 1 } , z _ { 2 } )$ with a carefully chosen random $D$ -dimensional approximation $R ( z _ { 1 } ) ^ { \\top } R ( z _ { 2 } )$ for $R ( z ) : \\mathbb { R } ^ { D _ { z } }  \\mathbb { R } ^ { D }$ , such that $\\begin{array} { r } { k ( z _ { 1 } , z _ { 2 } ) = \\mathbb { E } \\left[ R ( z _ { 1 } ) ^ { \\top } R ( z _ { 2 } ) \\right] , } \\end{array}$ . Fourier frequencies are sampled independently for the two kernels on $Z$ in $\\mathrm { H S I C } ( Z , Z )$ at each training step. We leave the details on how to construct $R ( z )$ for the kernels we use to Appendix C. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we present our experimental setup, where we assess the performance of the representation learned with SSL-HSIC both with and without a target network. First, we train a model with a standard ResNet-50 backbone using SSL-HSIC as objective on the training set of ImageNet ILSVRC-2012 [1]. For evaluation, we retain the backbone as a feature extractor for downstream tasks. We evaluate the representation on various downstream tasks including classification, object segmentation, object detection and depth estimation. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/a05cd99484a48004d410b54e0a447b170aabd0f27381d90ac49f4fcc73067ee3.jpg",
+        "image_caption": [
+            "Figure 4: Architecture and SSL-HSIC objective. A self-supervised label $y$ – an indicator of the image identity – is associated with an image $x$ . Image transformation functions $t$ are sampled and applied to the original image, resulting in views $t ^ { 1 } ( x )$ and $t ^ { 2 } ( x )$ . Features $z ^ { 1 }$ and $z ^ { 2 }$ are obtained after passing the augmented views through encoder $( f )$ , projector $( g )$ , and possibly predictor $( q )$ networks, while label $y$ is retained. Kernel matrices, $K$ for the latents and $L$ for the labels, are computed on the mini-batch of data; SSL-HSIC is estimated with $K$ and $L$ as in (12). The blue boxes reflect two potential options: when using a target network, $\\xi$ is a moving average of $\\theta$ , and a predictor network $q$ is added; without the target network, $q$ is removed and $\\xi$ is simply equal to $\\theta$ . "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Implementation ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Architecture Figure 4 illustrates the architecture we used for SSL-HSIC in this section. To facilitate comparison between different methods, our encoder $f _ { \\theta }$ uses the standard ResNet-50 backbone without the final classification layer. The output of the encoder is a 2048-dimension embedding vector, which is the representation used for downstream tasks. As in BYOL [25], our projector $g$ and predictor $q$ networks are 2-layer MLPs with 4096 hidden dimensions and 256 output dimensions. The outputs of the networks are batch-normalized and rescaled to unit norm before computing the loss. We use an inverse multiquadric kernel (IMQ) for the latent representation (approximated with 512 random Fourier features that are resampled at each step; see Appendix C for details) and a linear kernel for labels. $\\gamma$ in (4) is set to 3. When training without a target network, unlike SimSiam [24], we do not stop gradients for either branch. If the target network is used, its weights are an exponential moving average of the online network weights. We employ the same schedule as BYOL [25], $\\tau = 1 \\stackrel { - } { - } 0 . 0 1 \\stackrel { - } { \\cdot } ( \\cos ( \\pi t / T ) + 1 ) / 2$ with $t$ the current step and $T$ the total training steps. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Image augmentation Our method uses the same data augmentation scheme as BYOL (see Appendix D.1). Briefly, we first draw a random patch from the original image and resize it to $2 2 4 \\times 2 2 4$ Then, we apply a random horizontal flip, followed by color jittering, consisting of a random sequence of brightness, contrast, saturation, hue adjustments, and an optional grayscale conversion. Finally Gaussian blur and solarization are applied, and the view is normalized with ImageNet statistics. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Optimization We train the model with a batch size of 4096 on 128 Cloud TPU v4 cores. Again, following [19, 25], we use the LARS optimizer [52] with a cosine decay learning rate schedule over 1000 epochs. The base learning rate to all of our experiments is 0.4 and it is scaled linearly [53] with the batch size $l r = 0 . 4 \\times b a t c h { \\_ } s i z e / 2 5 6 .$ . All experiments use weight decay of $1 0 ^ { - 6 }$ . ",
+        "bbox": [
+            174,
+            794,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Learning kernel parameters We use a linear kernel for labels, since the type of kernel only scales (12). Our inverse multiquadric kernel for the latent $Z$ has an additional kernel scale parameter. We optimize this along with all other parameters, but regularize it to maximize the entropy of the distribution $k _ { \\sigma } ( s )$ , where $s _ { i j } = \\lVert { z } _ { i } - { z } _ { j } \\rVert ^ { 2 }$ ; this amounts to maximizing $\\log \\lVert k _ { \\sigma } ^ { \\prime } ( s ) \\rVert ^ { 2 }$ (Appendix D.1.2). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Linear evaluation on ImageNet Learned features are evaluated with the standard linear evaluation protocol commonly used in evaluating self-supervised learning methods [15–22, 25]. Table 1 reports the top-1 and top-5 accuracies obtained with SSL-HSIC on ImageNet validation set, and compares to previous self-supervised learning methods. Without a target network, our method reaches $7 2 . 2 \\%$ top-1 and $9 0 . 7 \\%$ top-5 accuracies. Unlike BYOL, the SSL-HSIC objective prevents the network from finding trivial solutions as explained in Section 3.2. Adding the target network, our method outperforms most previous methods, achieving top-1 accuracy of $7 4 . 8 \\%$ and top-5 accuracy of $9 2 . 2 \\%$ The fact that we see performance gains from adopting a target network suggests that its effect is not yet well understood, although note discussion in [25] which points to its stabilizing effect. ",
+        "bbox": [
+            173,
+            126,
+            825,
+            251
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/9816638d7e810e521d90e54d6e6ba7c6a00b6fc50c874d8aa8e0e255200af6ef.jpg",
+        "table_caption": [
+            "Table 1: Linear evaluation on the ImageNet validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Top-1(%)</td><td>Top-5(%)</td></tr><tr><td>Supervised [54]</td><td>75.9</td><td>92.8</td></tr><tr><td>SimCLR[19]</td><td>69.3</td><td>89.0</td></tr><tr><td>MoCo v2 [20]</td><td>71.1</td><td>90.1</td></tr><tr><td>BYOL [25]</td><td>74.3</td><td>91.6</td></tr><tr><td>SwAV[18]</td><td>75.3</td><td>-</td></tr><tr><td>Barlow Twins [22]</td><td>73.2</td><td>91.0</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>72.2</td><td>90.7</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>74.8</td><td>92.2</td></tr></table>",
+        "bbox": [
+            178,
+            292,
+            419,
+            405
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4ac8817fc095d16888065208da3e638cf45aa8a35425abf3cd6ab84a05206729.jpg",
+        "table_caption": [
+            "Table 2: Fine-tuning on $1 \\%$ , $10 \\%$ and $100 \\%$ of the ImageNet training set and evaluating on the validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Top-1(%)</td><td colspan=\"3\">Top-5(%)</td></tr><tr><td>1%</td><td>10%</td><td>100%</td><td>1%</td><td>10%</td><td>100%</td></tr><tr><td>Supervised [54]</td><td>25.4</td><td>56.4</td><td>75.9</td><td>48.4</td><td>80.4</td><td>92.8</td></tr><tr><td>SimCLR [19]</td><td>48.3</td><td>65.6</td><td>76.0</td><td>75.5</td><td>87.8</td><td>93.1</td></tr><tr><td>BYOL [25]</td><td>53.2</td><td>68.8</td><td>77.7</td><td>78.4</td><td>89.0</td><td>93.9</td></tr><tr><td>SwAV[18]</td><td>53.9</td><td>70.2</td><td>1</td><td>78.5</td><td>89.9</td><td>-</td></tr><tr><td>Barlow Twins [22]</td><td>55.0</td><td>69.7</td><td>-</td><td>79.2</td><td>89.3</td><td>-</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>45.3</td><td>65.5</td><td>76.4</td><td>72.7</td><td>87.5</td><td>93.2</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>52.1</td><td>67.9</td><td>77.2</td><td>77.7</td><td>88.6</td><td>93.6</td></tr></table>",
+        "bbox": [
+            450,
+            292,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Semi-supervised learning on ImageNet We fine-tune the network pretrained with SSL-HSIC on $1 \\%$ , $10 \\%$ and $100 \\%$ of ImageNet, using the same ImageNet splits as SimCLR [19]. Table 2 summarizes the semi-supervised learning performance. Our method, with or without a target network, has competitive performance in both data regimes. The target network has the most impact on the small-data regime, with $1 \\%$ labels. ",
+        "bbox": [
+            173,
+            419,
+            826,
+            488
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/7416842a8e690375d9461297698e31842b7ef2c559bf849eb6a9796726edc4e0.jpg",
+        "table_caption": [
+            "Table 3: Comparison of transfer learning performance on 12 image datasets. Supervised-IN is trained on ImageNet with supervised pretrainining. Random init trains on individual dataset with randomly initialized weights. MPCA refers to mean per-class accuracy; AP50 is average precision at $\\mathrm { I o U } { = } 0 . 5$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset Metric</td><td>Birdsnap Caltechl01 Cifarl0 Cifarl001 Top-1</td><td>MPCA</td><td></td><td>Top-1</td><td>DTD Top-1</td><td>Aircraft Food Flowers</td><td>MPCA Top-1</td><td>MPCA MPCA Top-1</td><td>Pets</td><td>Cars</td><td>Top-1</td><td>SUN397 VOC2007 AP50</td></tr><tr><td>Linear:</td><td></td><td></td><td>Top-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Supervised-IN [19]</td><td>53.7</td><td>94.5</td><td>93.6</td><td>78.3</td><td></td><td>61.0</td><td>72.3</td><td>94.7</td><td>91.5</td><td>67.8</td><td>61.9</td><td>82.8</td></tr><tr><td>SimCLR[19]</td><td>37.4</td><td>90.3</td><td>90.6</td><td>71.6</td><td>74.9 74.5</td><td>50.3</td><td>68.4</td><td>90.3</td><td>83.6</td><td>50.3</td><td>58.8</td><td>80.5</td></tr><tr><td>BYOL [25]</td><td>57.2</td><td>94.2</td><td>91.3</td><td>78.4</td><td>75.5</td><td>60.6</td><td>75.3</td><td>96.1</td><td>90.4</td><td>66.7</td><td>62.2</td><td>82.5</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>50.6</td><td>92.3</td><td>91.5</td><td>75.9</td><td>75.3</td><td>57.9</td><td>73.6</td><td>95.0</td><td>88.2</td><td>59.3</td><td>61.0</td><td>81.4</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>57.8</td><td>93.5</td><td>92.3</td><td>77.0</td><td>76.2</td><td>58.5</td><td>75.6</td><td>95.4</td><td>91.2</td><td>62.6</td><td>61.8</td><td>83.3</td></tr><tr><td>Fine-tuned:</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><tr><td>Supervised-IN [19]</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>85.0</td></tr><tr><td>Random init [19]</td><td>75.8</td><td>93.3</td><td>97.5</td><td>86.4</td><td>74.6</td><td>86.0 85.9</td><td>88.3</td><td>97.6</td><td>92.1</td><td>92.1</td><td>94.3 53.6</td><td>67.3</td></tr><tr><td>SimCLR[19]</td><td>76.1 75.9</td><td>72.6 92.1</td><td>95.9 97.7</td><td>80.2 85.9</td><td>64.8 73.2</td><td>88.1</td><td>86.9 88.2</td><td>92.0 97.0</td><td>81.5 89.2</td><td>91.4 91.3</td><td>63.5</td><td>84.1</td></tr><tr><td>BYOL [25]</td><td>76.3</td><td>93.8</td><td>97.8</td><td>86.1</td><td>76.2</td><td>88.1</td><td>88.5</td><td>97.0</td><td>91.7</td><td>91.6</td><td>63.7</td><td>85.4</td></tr><tr><td>SSL-HSIC (w/o target)</td><td>73.1</td><td>91.5</td><td>97.4</td><td>85.3</td><td>75.3</td><td>87.1</td><td>87.5</td><td>96.4</td><td>90.6</td><td>91.6</td><td>62.2</td><td>84.1</td></tr><tr><td>SSL-HSIC (w/ target)</td><td>74.9</td><td>93.8</td><td>97.8</td><td>84.7</td><td>75.4</td><td>88.9</td><td>87.7</td><td>97.3</td><td>91.7</td><td>91.8</td><td>61.7</td><td>84.1</td></tr></table>",
+        "bbox": [
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+            542,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Transfer to other classification tasks To investigate the generality of the representation learned with SSL-HSIC, we evaluate the transfer performance for classification on 12 natural image datasets [55–64] using the same procedure as [19, 25, 65]. Table 3 shows the top-1 accuracy of the linear evaluation and fine-tuning performance on the test set. SSL-HSIC gets state-of-the-art performance on 3 of the classification tasks and reaches strong performance on others for this benchmark, indicating the learned representations are robust for transfer learning. ",
+        "bbox": [
+            173,
+            738,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Transfer to other vision tasks To test the ability of transferring to tasks other than classification, we fine-tune the network on semantic segmentation, depth estimation and object detection tasks. We use Pascal VOC2012 dataset [58] for semantic segmentation, NYU v2 dataset [66] for depth estimation and COCO [67] for object detection. Object detection outputs either bounding box or object segmentation (instance segmentation). Details of the evaluations setup is in Appendix D.2. Table 4 and Table 5 shows that SSL-HSIC achieves competitive performance on all three vision tasks. ",
+        "bbox": [
+            174,
+            827,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b80a57a29e2a954c8afa620a36f1ce609b0a2d5f2fe67f0a2d5fb45bac988fa.jpg",
+        "table_caption": [
+            "Table 4: Fine-tuning performance on semantic segmentation and depth estimation. Mean Intersection over Union (mIoU) is reported for semantic segmentation. Relative error (rel), root mean squared error (rms), and the percent of pixels (pct) where the error is below $1 . 2 5 ^ { n }$ thresholds are reported for depth estimation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">VOC2012 mIoU</td><td colspan=\"5\">NYU v2</td></tr><tr><td>pct.&lt;1.25</td><td>pct.&lt;1.252</td><td>pct.&lt;1.253</td><td>rms</td><td>rel</td></tr><tr><td>Supervised-IN</td><td>74.4</td><td>81.1</td><td>95.3</td><td>98.8</td><td>0.573</td><td>0.127</td></tr><tr><td>SimCLR</td><td>75.2</td><td>83.3</td><td>96.5</td><td>99.1</td><td>0.557</td><td>0.134</td></tr><tr><td>BYOL</td><td>76.3</td><td>84.6</td><td>96.7</td><td>99.1</td><td>0.541</td><td>0.129</td></tr><tr><td>SSL-HSIC(w/o target)</td><td>74.9</td><td>84.1</td><td>96.7</td><td>99.2</td><td>0.539</td><td>0.130</td></tr><tr><td>SSL-HSIC(w/ target)</td><td>76.0</td><td>83.8</td><td>96.8</td><td>99.1</td><td>0.548</td><td>0.130</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/4667c209a2ad0485966510d9de070969285cc2c2254eda2bae9903893ef03582.jpg",
+        "table_caption": [
+            "Table 5: Fine-tuning performance on COCO object detection tasks. Precision, averaged over $1 0 \\ \\mathrm { I o U }$ (Intersection over Union) thresholds, is reported for both bounding box and object segmentation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>APbb</td><td>Apmk</td></tr><tr><td>Supervised</td><td>39.6</td><td>35.6</td></tr><tr><td>SimCLR</td><td>39.7</td><td>35.8</td></tr><tr><td>MoCo v2</td><td>40.1</td><td>36.3</td></tr><tr><td>BYOL</td><td>41.6</td><td>37.2</td></tr><tr><td>SwAV</td><td>41.6</td><td>37.8</td></tr><tr><td>SSL-HSIC(w/o target)</td><td>40.5</td><td>36.3</td></tr><tr><td>SSL-HSIC(w/ target)</td><td>41.3</td><td>36.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 Ablation Studies ",
+        "text_level": 1,
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+        "text": "We present ablation studies to gain more intuition on SSL-HSIC. Here, we use a ResNet-50 backbone trained for 100 epochs on ImageNet, and evaluate with the linear protocol unless specified. ",
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+        "type": "text",
+        "text": "ResNet architectures In this ablation, we investigate the performance of SSL-HSIC with wider and deeper ResNet architecture. Figure 1 and Table 6 show our main results. The performance of SSL-HSIC gets better with larger networks. We used the supervised baseline from [25] which our training framework is based on ([19] reports lower performance). The performance gap between SSL-HSIC and the supervised baseline diminishes with larger architectures. In addition, Table 7 presents the semi-supervise learning results with subsets $1 \\%$ , $10 \\%$ and $100 \\%$ of the ImageNet data. ",
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+    {
+        "type": "table",
+        "img_path": "images/0680ab887cfe1d473b62aa0681b0535929e6999e84b4a515040d1aeb5f7c6969.jpg",
+        "table_caption": [
+            "Table 6: Top-1 and top-5 accuracies for different ResNet architectures using linear evaluation protocol. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">SSL-HSIC</td><td colspan=\"2\">BYOL[25]</td><td colspan=\"2\">Sup.[25]</td></tr><tr><td>ResNet</td><td>Top1</td><td>Top5</td><td>Top1</td><td>Top5</td><td>Top1</td><td>Top5</td></tr><tr><td>50 (1x)</td><td>74.8</td><td>92.2</td><td>74.3</td><td>91.6</td><td>76.4</td><td>92.9</td></tr><tr><td>50 (2x)</td><td>77.9</td><td>94.0</td><td>77.4</td><td>93.6</td><td>79.9</td><td>95.0</td></tr><tr><td>50(4x)</td><td>79.1</td><td>94.5</td><td>78.6</td><td>94.2</td><td>80.7</td><td>95.3</td></tr><tr><td>200 (2x)</td><td>79.6</td><td>94.8</td><td>79.6</td><td>94.9</td><td>80.1</td><td>95.2</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Table 7: Top-1 and top-5 accuracies for different ResNet architectures using semisupervised fine-tuning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Top1</td><td colspan=\"3\">Top5</td></tr><tr><td>ResNet</td><td>1%</td><td>10%</td><td>100%</td><td>1%</td><td>10%</td><td>100%</td></tr><tr><td>50 (1x)</td><td>52.1</td><td>67.9</td><td>77.2</td><td>77.7</td><td>88.6</td><td>93.6</td></tr><tr><td>50 (2x)</td><td>61.2</td><td>72.6</td><td>79.3</td><td>83.8</td><td>91.2</td><td>94.7</td></tr><tr><td>50 (4x)</td><td>67.0</td><td>75.4</td><td>79.7</td><td>87.4</td><td>92.5</td><td>94.8</td></tr><tr><td>200(2x)</td><td>69.0</td><td>76.3</td><td>80.5</td><td>88.3</td><td>92.9</td><td>95.2</td></tr></table>",
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+        "type": "text",
+        "text": "Regularization term We compare performance of InfoNCE with SSL-HSIC in Table 8a since they can be seen as approximating the same $\\mathrm { H S I C } ( Z , Y )$ objective but with different forms of regularization. We reproduce the InfoNCE result in our codebase, using the same architecture and data augmentiation as for SSL-HSIC. Trained for 100 epochs (without a target network), InfoNCE achieves $6 6 . 0 \\%$ top-1 and $8 6 . 9 \\%$ top-5 accuracies, which is better than the result reported in [19]. For comparison, SSL-HSIC reaches $6 6 . 7 \\%$ top-1 and $8 7 . 6 \\%$ top-5 accuracies. This suggests that the regularization employed by SSL-HSIC is more effective. ",
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+        "type": "text",
+        "text": "Kernel type We investigate the effect of using different a kernel on latents $Z$ . Training without a target network or random Fourier feature approximation, the top-1 accuracies for linear, Gaussian, and inverse multiquadric (IMQ) kernels are $6 5 . 2 7 \\%$ , $6 6 . 6 7 \\%$ and $6 6 . 7 2 \\%$ respectively. Non-linear kernels indeed improve the performance; Gaussian and IMQ kernels reach very similar performance for 100 epochs. We choose IMQ kernel for longer runs, because its heavy-tail property can capture more signal when points are far apart. ",
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+    {
+        "type": "text",
+        "text": "Number of RFF Features Table 8b shows the performance of SSL-HSIC with different numbers of Fourier features. The RFF approximation has a minor impact on the overall performance, as long as we resample them; fixed sets of features performed poorly. Our main result picked 512 features, for substantial computational savings with minor loss in accuracy. ",
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+    {
+        "type": "text",
+        "text": "Batch size Similar to most of the self-supervised learning methods [19, 25], SSL-HSIC benefits from using a larger batch size during training. However, the drop of performance from using smaller batch size is not as pronounced as it is in SimCLR[19] as shown in Table 8c. ",
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+    {
+        "type": "table",
+        "img_path": "images/9035c053c539fa3c75259cd473bf307a2eeffeb6c29d7088e0a836195d641218.jpg",
+        "table_caption": [
+            "(a) Regularization "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Top-1</td><td>Top-5</td></tr><tr><td>SSL-HSIC</td><td>66.7</td><td>87.6</td></tr><tr><td>InfoNCE</td><td>66.0</td><td>86.9</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/35caf24f4a2c86fccd84be89c5db0321f33bf20deaa7fd070e4337c8837c3314.jpg",
+        "table_caption": [
+            "Table 8: Linear evaluation results when varying different hyperparameters. ",
+            "(d) Projector/predictor size "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Output Dim</td><td>Top-1(%)</td></tr><tr><td>64</td><td>65.4</td></tr><tr><td>128</td><td>66.0</td></tr><tr><td>256</td><td>66.4</td></tr><tr><td>512</td><td>66.6</td></tr><tr><td>1024</td><td>66.6</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/1d5b2f05662825431d99f28568973bde10e24f890522e49407153fb772150317.jpg",
+        "table_caption": [
+            "(b) # Fourier features "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#RFFs</td><td>Top-1(%)</td></tr><tr><td>64</td><td>66.0</td></tr><tr><td>128</td><td>66.2</td></tr><tr><td>256</td><td>66.2</td></tr><tr><td>512</td><td>66.4</td></tr><tr><td>1024</td><td>66.5</td></tr><tr><td>2048</td><td>66.5</td></tr><tr><td>No Approx.</td><td>66.7</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f748bd0018c5619e5b3b530a607f0d56d93a2079015f5736d08550c3905c2787.jpg",
+        "table_caption": [
+            "(c) Batch size "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"4\">Top-1(%)</td></tr><tr><td>Batch Size</td><td>SSL-HSIC</td><td>SimCLR</td><td></td></tr><tr><td>256</td><td>63.7</td><td>57.5</td><td></td></tr><tr><td>512</td><td>65.6</td><td></td><td>60.7</td></tr><tr><td>1024</td><td>66.7</td><td></td><td>62.8</td></tr><tr><td>2048</td><td>67.1</td><td></td><td>64.0</td></tr><tr><td>4096</td><td>66.7</td><td></td><td>64.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Projector and predictor output size Table 8d shows the performance when using different output dimension for the projector/predictor networks. The performance saturates at 512 dimensions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We introduced SSL-HSIC, a loss function for self-supervised representation learning based on kernel dependence maximization. We provided a unified view on various self-supervised learning losses: we proved that InfoNCE, a lower bound of mutual information, actually approximates SSL-HSIC with a variance-based regularization, and we can also interpret SSL-HSIC as metric learning where the cluster structure is imposed by the self-supervised label, of which the BYOL objective is a special case. We showed that training with SSL-HSIC achieves performance on par with the state-of-the-art on the standard self-supervised benchmarks. ",
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+    {
+        "type": "text",
+        "text": "Although using the image identity as self-supervised label provides a good inductive bias, it might not be wholly satisfactory; we expect that some images pairs are in fact more similar than others, based e.g. on their ImageNet class label. It will be interesting to explore methods that combine label structure discovery with representation learning (as in SwAV [18]). In this paper, we only explored learning image representations, but in future work SSL-HSIC can be extended to learning structure for $Y$ as well, building on existing work [41, 45]. ",
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+    {
+        "type": "text",
+        "text": "Broader impact ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our work concentrates on providing a more theoretically grounded and interpretable loss function for self-supervised learning. A better understanding of self-supervised learning, especially through more interpretable learning dynamics, is likely to lead to better and more explicit control over societal biases of these algorithms. SSL-HSIC yields an alternative, clearer understanding of existing selfsupervised methods. As such, it is unlikely that our method introduces further biases than those already present for self-supervised learning. ",
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+    {
+        "type": "text",
+        "text": "The broader impacts of the self-supervised learning framework is an area that has not been studied by the AI ethics community, but we think it calls for closer inspection. An important concern for fairness of ML algorithms is dataset bias. ImageNet is known for a number of problems such as offensive annotations, non-visual concepts and lack of diversity, in particular for underrepresented groups. Existing works and remedies typically focus on label bias. Since SSL doesn’t use labels, however, the type and degree of bias could be very different from that of supervised learning. To mitigate the risk of dataset bias, one could employ dataset re-balancing to correct sampling bias [68] or completely exclude human images from the dataset while achieving the same performance [69]. ",
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+    {
+        "type": "text",
+        "text": "A new topic to investigate for self-supervised learning is how the bias/unbiased representation could be transferred to downstream tasks. We are not aware of any work in this direction. Another area of concern is security and robustness. Compared to supervised learning, self-supervised learning typically involves more intensive data augmentation such as color jittering, brightness adjustment, etc. There is some initial evidence suggesting self-supervised learning improves model robustness [70]. However, since data augmentation can either be beneficial [71] or detrimental [72] depending on the type of adversarial attacks, more studies are needed to assess its role for self-supervised learning. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The authors would like to thank Olivier J. Hénaff for valuable feedback on the manuscript and the help for evaluating object detection task. We thank Aaron Van den Oord and Oriol Vinyals for providing valuable feedback on the manuscript. We are grateful to Yonglong Tian, Ting Chen and the BYOL authors for the help with reproducing baselines and evaluating downstream tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by DeepMind, the Gatsby Charitable Foundation, the Wellcome Trust, NSERC, and the Canada CIFAR AI Chairs program. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
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+    },
+    {
+        "type": "text",
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Le. “Do Better ImageNet Models Transfer Better?” CVPR. 2019. arXiv: 1805.08974.   \n[66] N. Silberman, D. Hoiem, P. Kohli, and R. Fergus. “Indoor Segmentation and Support Inference from RGBD Images.” ECCV. 2012.   \n[67] T.-Y. Lin, M. Maire, S. J. Belongie, L. D. Bourdev, R. B. Girshick, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick. “Microsoft COCO: Common Objects in Context.” ECCV. 2014. arXiv: 1405.0312.   \n[68] K. Yang, K. Qinami, L. Fei-Fei, J. Deng, and O. Russakovsky. “Towards fairer datasets: Filtering and balancing the distribution of the people subtree in the imagenet hierarchy.” Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency. 2020, pp. 547–558.   \n[69] Y. Asano, C. Rupprecht, A. Zisserman, and A. Vedaldi. “PASS: An ImageNet replacement for self-supervised pretraining without humans” (2021).   \n[70] D. Hendrycks, M. Mazeika, S. Kadavath, and D. Song. “Using self-supervised learning can improve model robustness and uncertainty.” arXiv preprint arXiv:1906.12340 (2019).   \n[71] A. Sablayrolles, M. Douze, C. Schmid, Y. Ollivier, and H. Jégou. “White-box vs black-box: Bayes optimal strategies for membership inference.” International Conference on Machine Learning. PMLR. 2019, pp. 5558–5567.   \n[72] D. Yu, H. Zhang, W. Chen, J. Yin, and T.-Y. Liu. “How Does Data Augmentation Affect Privacy in Machine Learning?” Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 35. 12. 2021, pp. 10746–10753.   \n[73] L. Grafakos. Classical Fourier Analysis. Vol. 2. Springer, 2008.   \n[74] R. Piessens. “The Hankel transform.” The Transforms and Applications Handbook 2.9 (2000).   \n[75] F. Johansson et al. mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.18). http://mpmath.org/. 2013.   \n[76] J. Long, E. Shelhamer, and T. Darrell. “Fully Convolutional Networks for Semantic Segmentation.” CVPR. 2015. arXiv: 1411.4038.   \n[77] O. J. Hénaff, S. Koppula, J.-B. Alayrac, A. van den Oord, O. Vinyals, and J. Carreira. “Efficient Visual Pretraining with Contrastive Detection.” 2021. arXiv: 2103.10957.   \n[78] K. He, G. Gkioxari, P. Dollár, and R. B. Girshick. “Mask R-CNN.” ICCV. 2017. arXiv: 1703.06870.   \n[79] T.-Y. Lin, P. Dollár, R. B. Girshick, K. He, B. Hariharan, and S. J. Belongie. “Feature Pyramid Networks for Object Detection.” CVPR. 2017. arXiv: 1612.03144. ",
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parse/train/3ccoZ40Us0N/3ccoZ40Us0N.md

@@ -0,0 +1,224 @@
+# CLIP-It! Language-Guided Video Summarization
+
+Medhini Narasimhan Anna Rohrbach Trevor Darrell
+
+University of California, Berkeley {medhini, anna.rohrbach, trevordarrell}@berkeley.edu https://medhini.github.io/clip_it
+
+![](images/d7d642572e87e426a8435e58d17586d42b74fd301a04f4a0c71e972de49e802f.jpg)  
+Figure 1: We introduce CLIP-It, a language-guided multimodal transformer for generic and query-focused video summarization. The figure shows results from our method. Given a day-long video of a national park tour, the generic summary (top) is a video with relevant and diverse keyframes. When using the query “All the scenes containing restaurants and shopping centers”, the generated query-focused summary includes all the matching scenes. Similarly, the query “All water bodies such as lakes, rivers, and waterfalls”, yields a short summary containing all the water bodies present in the video.
+
+# Abstract
+
+A generic video summary is an abridged version of a video that conveys the whole story and features the most important scenes. Yet the importance of scenes in a video is often subjective, and users should have the option of customizing the summary by using natural language to specify what is important to them. Further, existing models for fully automatic generic summarization have not exploited available language models, which can serve as an effective prior for saliency. This work introduces CLIP-It, a single framework for addressing both generic and queryfocused video summarization, typically approached separately in the literature. We propose a language-guided multimodal transformer that learns to score frames in a video based on their importance relative to one another and their correlation with a user-defined query (for query-focused summarization) or an automatically generated dense video caption (for generic video summarization). Our model can be extended to the unsupervised setting by training without ground-truth supervision. We outperform baselines and prior work by a significant margin on both standard video summarization datasets (TVSum and SumMe) and a query-focused video summarization dataset (QFVS). Particularly, we achieve large improvements in the transfer setting, attesting to our method’s strong generalization capabilities.
+
+# 1 Introduction
+
+An effective video summary captures the essence of the video and provides a quick overview as an alternative to viewing the whole video; it should be succinct yet representative of the entire video. Summarizing videos has many use cases - for example, viewers on YouTube may want to watch a short summary of the video to assess its relevance. While a generic summary is useful for capturing the important scenes in a video, it is even more practical if the summary can be customized by the user. As seen in Fig. 1, users should be able to indicate the concepts of the video they would like to see in the summary using natural language queries.
+
+Generic video summarization datasets such as TVSum [36] and SumMe [7] provide ground-truth annotations in the form of frame or shot-level importance scores specified by multiple annotators. Several learning-based methods reduce the task to a frame-wise score prediction problem. Sequence labeling methods [5, 42, 43, 44, 18] model variable-range dependencies between frames but fail to capture relationships across all frames simultaneously. While attention [3] and graph based [26] methods address this partly, they disregard ordering of the input frames which is useful when predicting scores. Moreover, these methods use only visual cues to produce the summaries and cannot be customized with a natural language input. Another line of work, query-focused video summarization [32], allows the users to customize the summary by specifying a query containing two concepts (eg., food and drinks). However, in their work the query can only be chosen from a fixed set of predefined concepts which limits the flexibility for the user.
+
+It is important to note that efforts in generic and query-focused video summarization have so far been disjoint, with no single method for both. Our key innovation is to unify these tasks in one language-guided framework. We introduce CLIP-It, a multimodal summarization model which takes two inputs, a video and a natural language text, and synthesizes a summary video conditioned on the text. In case of generic video summarization, the natural language text is a video description obtained using an off-the-shelf dense video captioning method. Alternatively, in the case of query-focused video summarization, the language input is a user-defined query. Unlike existing generic methods which only use visual cues, we show that adding language as an input leads to the “discovery” of relevant concepts and actions resulting in better summaries. Our method uses a Transformer with positional encoding, which can attend to all frames at once (unlike LSTM based methods) and also keep track of the ordering of frames (unlike graph based methods). In contrast to existing query-focused methods [32] which only allow keyword based queries, we aim to enable open-ended natural language queries for users to customize video summaries. For example, as seen in Fig. 1, using our method users can specify long and descriptive queries such as, “All water bodies such as lakes, rivers, and waterfalls” which is not possible with previous methods.
+
+Given an input video, CLIP-It generates a video summary guided by either a user-defined natural language query or a system generated description. It uses a Language-Guided Attention head to compute a joint representation of the image and language embeddings, and a Frame-Scoring Transformer to assign scores to individual frames in the video using the fused representations. Following [43, 42], the summary video is constructed from high scoring frames by converting framelevel scores to shot-level scores and using knapsack algorithm to fit the maximum number of high scoring shots in a timed window. Fig. 1 shows an output from our method. Given an hour long video of a national park tour, we generate a 2 minute generic summary comprising of all the important scenes in the video. Given the two language queries, our method picks the matching keyframes in both cases. We can train CLIP-It without ground-truth supervision by leveraging the reconstruction and diversity losses [9, 31]. For generic video summarization, we evaluate our approach on the standard benchmarks, TVSum and SumMe. We achieve performance improvement on F1 score of nearly $3 \%$ in the supervised setting and $4 \%$ in the unsupervised setting on both datasets. We show large gains $( 5 \% )$ in the Transfer setting, where CLIP-It is trained and evaluated on disjoint sets of data. For the queryfocused scenario we evaluate on the QFVS dataset [33], where we also achieve state-of-the-art results.
+
+To summarize our contributions, we introduce CLIP-It, a language-guided model that unifies generic and query-focused video summarization. Our approach uses language conditioning in the form of off-the-shelf video descriptions (for generic summarization) or user-defined natural language queries (for query-focused summarization). We show that the Transformer design enables effective contextualization across frames, benefiting our tasks. We also demonstrate the impact of language guidance on generic summarization. Finally, we establish the new state-of-the-art on both generic and query-focused datasets in supervised and unsupervised settings.
+
+# 2 Related Work
+
+Generic Video Summarization. A video summary is a short synopsis created by stitching together important clips from the original video. Early works [21, 35, 34] referred to it as Video Skimming or Dynamic Video Summarization and used hand-designed features to generate summaries. Likewise, non-parametric unsupervised methods [13, 16, 23, 19, 20, 28] used various heuristics to represent the importance of frames. Introduction of benchmark datasets such as TVSum [36] and SumMe [7] provided relevance scores for frames in videos annotated by users, resulting in multiple human generated summaries. This enabled automatic evaluation of video summarization techniques and has lead to the development of many supervised learning based methods [6, 8, 22, 26, 31, 30, 41, 42, 43, 44, 45]. These approaches capture high-level semantic information and outperform the heuristic unsupervised methods. Fully convolutional sequence networks [31] treat video summarization as a binary label prediction problem. Determinantal point processes [15] and LSTM [10] based approaches [5, 18, 42, 43, 44] model variable-range dependencies between frames. However, these are sequential and fail to capture relationships across all frames simultaneously. Attention [3] and graph based methods [26] address this issue by modeling relationships across all frames, but they disregard the ordering of frames in a video, which is also useful for summarization. Our method uses a Transformer [38] with positional encoding, which allows for joint attention across all frames while maintaining an ordering of the input sequence of frames.
+
+Some of the above works have supervised and unsupervised variants with modifications to the objective function. Specifically, for the unsupervised variant, they use reconstruction and diversity losses which do not require ground truth. We follow prior work in terms of using the same loss functions. Other notable unsupervised approaches include an adversarial LSTM based method [22], a generative adversarial network to learn from unpaired data [30], and a cycle consistent learning objective [41].
+
+Video-Text Summarization. Plummer et al. [27] use image-language embeddings for generating video summaries but evaluate their approach in the text domain, and not on the generic video summarization benchmarks. Furthermore, they require text to be provided as input and don’t have a mechanism to generate captions if absent. On the other hand, our method works well with off-theshelf captions and we evaluate on both generic and query-focused benchmark datasets. Chen et al. [1] jointly train a text and video summarization network but rely on ground-truth text summaries.
+
+Query-Focused Video Summarization. Oftentimes users browsing videos on YouTube are looking for something specific so a generic summary might not suffice. In this case, there should be an option to customize the generated summary using a natural language query. Sharghi et al. [32] introduce the Query-Focused Video Summarization (QFVS) dataset for UT Egocentric [16] videos containing user-defined video summaries for a set of pre-defined concepts. Sharghi et al. [33] propose a memory network to attend over different video frames and shots with the user query as input. However, this recurrent attention mechanism precludes parallelization and limits modeling long-range dependencies, which is overcome by our Transformer architecture. Moreover, their method only works with the pre-defined set of keyword based queries in QFVS dataset. Since we use CLIP [29] to encode the language input and train our method on dense video descriptions, this allows users to define freeform queries at test time (as seen in Fig. 1). Other works [12, 39] similarly condition the summary generation on keyword based queries but haven’t released their data.
+
+Transformers. Transformers [38] were introduced for neural machine translation and have since been applied to video-language tasks such as video-retrieval [4] and video captioning [17, 48]. In this work we adapt transformers for video summarization. We modify self-attention [38] to a LanguageGuided Attention block that accepts inputs from two modalities. Additionally, our method also relies on CLIP [29] for extracting image and text features and a Bi-Modal Transformer [11] for dense video caption generation, both of which also have transformer backbone.
+
+# 3 CLIP-It: Language-Guided Video Summarization
+
+CLIP-It is a unified language-guided framework for both generic and query-focused video summarization. It takes an input video and a user-defined query or a system generated dense video caption and constructs a summary by selecting key frames from the video. First, we explain the intuition behind our approach. In the case of query-focused summarization, clearly, it is necessary to model the user query as input in order to produce an appropriate summary. In the case of generic video summarization no user query is available; nonetheless, we show here that we can leverage the semantic information contained in associated natural language descriptions to guide the summarization process. Assuming we had a generic description that accompanies a video (e.g., A person is walking a dog. The person throws a ball. The dog runs after it.), we could leverage its semantic embedding to match $i t$ to the most relevant portions of the video. Such a description could be obtained automatically by generating dense video captions [11]. We first present an overview of our approach, CLIP-It, followed by a detailed description of the individual components.
+
+![](images/5cf0ec307f855e688d0cd7ddc67c06c5416e4e5a59a1eb7b6e6c867ced503bf9.jpg)  
+Figure 2: Overview of CLIP-It. Given an input video, CLIP-It generates a summary conditioned on either a user-defined natural language query or an automatically generated dense video caption. The Language-Guided Attention head fuses the image and language embeddings and the Frame-Scoring Transformer jointly attends to all frames to predict their relevance scores. During inference, the video summary is constructed by converting frame scores to shot scores and using Knapsack algorithm to select high scoring shots.
+
+Overview. Our approach, CLIP-It, is outlined in $\mathrm { F i g } 2$ . Given a video, we extract $N$ frames denoted by $F _ { i }$ , $i \in [ 1 , \dots \bar { N } ]$ . We formulate the video summarization task as a per-frame binary classification problem. We embed the frames using a pretrained network $f _ { i m g }$ . If a query is provided (in the form of a natural language string), we embed the query using a pretrained network $f _ { t x t }$ . Alternatively, as seen in the figure, we use an off-the-shelf video captioning model to generate a dense video caption with $M$ sentences denoted by $C _ { j }$ , $j \in [ 1 , \bar { \dots { M } } ]$ and embed each sentence using the pretrained network $f _ { t x t }$ . Next, we compute language attended image embeddings $I ^ { * }$ using learned Language-Guided Multi-head Attention $f _ { i m g \_ t x t } ^ { * }$ . Finally, we train a Frame-Scoring Transformer which assigns scores to each frame in the video (green indicating a high score and red indicating a low score). To construct the video summary during inference, we convert frame-level scores to shot-level scores and finally, use 0/1 knapsack algorithm to pick the key shots [43]. In the following, we describe the Language-Guided Attention and the Frame-Selection Transformer modules, followed by discussing the visual and language encoders.
+
+Language-Guided Attention. We use Language-Guided Multi-head Attention to efficiently fuse information across the video and language modalities and to infer long-term dependencies across both. Using a single attention head does not suffice as our goal is to allow all captions to attend to all frames in the video. We modify the Multi-Head Attention described in Vaswani et al. [38] to take in inputs from both modalities. We set Query $Q$ , Key $K$ , and Value $V$ as follows,
+
+$$
+\begin{array} { c } { { \therefore , \nu \ - \jmath \iota x \iota \mathrm { ( } \mathrm { V } _ { \mathcal { I } } \mathrm { ) , ~ } \mathrm { w a n g e - } \mathrm { G u i d e d ~ A t t n . } ( Q , K , V ) = \mathrm { C o n c a t ( h e a d _ 1 , } . . . , \mathrm { h e a d _ h } ) W ^ { O } , } } \\ { { \mathrm { L a n g u a g e - } \mathrm { G u i d e d ~ A t t n . } ( Q , K , V ) = \mathrm { C o n c a t ( h e a d _ 1 , } . . . , \mathrm { h e a d _ h } ) W ^ { O } , } } \\ { { \mathrm { w h e r e ~ h e a d _ i = A t t e n t i o n } ( Q W _ { i } ^ { Q } , K W _ { i } ^ { K } , V W _ { i } ^ { V } ) } } \\ { { \mathrm { a n d ~ A t t e n t i o n } ( Q , K , V ) = \mathrm { s o f m a x } ( \displaystyle \frac { Q K ^ { T } } { \sqrt { d _ { k } } } ) V } } \end{array}
+$$
+
+$\it { W _ { i } ^ { Q } }$ , $W _ { i } ^ { K }$ , and $W _ { i } ^ { V }$ are learned parameter matrices and $d _ { k }$ is the dimensions of $K$ . The output of the Language-Guided Multi-Head Attention are the attended image embeddings, denoted as $F _ { i } ^ { ' }$ .
+
+Frame-Scoring Transformer. Finally, it is also important to ensure that we do not include redundant information, e.g., several key shots from the same event. To better model interactions across frames and contextualize them w.r.t. each other, we add a Frame-Scoring Transformer that takes image-text representations as input and outputs one score per frame. Based on the Transformer model [38], this module assigns relevance scores to the attended image embeddings $F _ { i } ^ { ' }$ . We feed $F _ { i } ^ { ' }$ to the bottom of both the encoder and decoder stacks. Similar to [38], we use positional encoding to insert information about the relative positions of the tokens in the sequence. We add positional encodings to the input embeddings at the bottom of the encoder and decoder stacks.
+
+Image Encoding. We encode the image using a pre-trained network $f _ { i m g }$ . We experiment with the following networks: GoogleNet [37] (for a fair comparison to prior work), ResNet [8], and CLIP [29].
+
+Text Encoding. We encode the user-defined query or the system generated dense caption using a pre-trained network $f _ { t x t }$ . In this case we tried the CLIP (ViT and RN101) model. In case of generic video summarization, we generate dense video captions for the whole video in order to condition on language and incorporate added semantics into our video summarization pipeline. We use the Bi-Modal Transformer [11] dense video captioning model that generates a multi-sentence description given a video with audio. Since there are multiple sentences in the dense video caption, we first embed each sentence of the caption using the text encoder $f _ { t x t }$ described above. They are then concatenated and fused using a multi-layer perceptron (MLP).
+
+# 3.1 Learning
+
+We employ 3 loss functions (classification, diversity, and reconstruction) to train our model. The supervised setting uses all 3 and the unsupervised setting uses only diversity and reconstruction losses.
+
+Classification loss. We use a weighted binary cross entropy loss for classifying each frame:
+
+$$
+\mathcal { L } _ { c } = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } w ^ { * } [ x _ { i } ^ { * } \mathrm { l o g } ( x _ { i } ) ] + ( 1 - w ^ { * } ) [ ( 1 - x _ { i } ^ { * } ) \mathrm { l o g } ( 1 - x _ { i } ) ] ,
+$$
+
+where $\boldsymbol { x } _ { i } ^ { * }$ is the ground-truth label of the $i$ -th frame and $N$ is the total number of frames in the $w ^ { * }$ weif ht assigned to the class is a background frame $\boldsymbol { x } _ { i } ^ { * }$ , which is set to $\frac { \# k e y f r a m e s } { N }$ if $\boldsymbol { x } _ { i } ^ { * }$ is a keyframe and $1 - { \frac { \# k e y f r a m e s } { N } }$ $\boldsymbol { x } _ { i } ^ { * }$
+
+For the purpose of training without any supervision, we employ two additional losses that enforce diversity in the selected keyframes. We first select keyframes $X$ based on the scores assigned by the Frame-Scoring Transformer. We pass the output features of the transformer model for the selected keyframes through a decoder network consisting of $1 \times 1$ convolution layers to obtain reconstructed feature vectors for the selected keyframes such that each keyframe feature vector is of the same dimension as its corresponding input frame-level feature vector.
+
+Reconstruction Loss. The reconstruction loss $\mathcal { L } _ { r }$ is defined as the mean squared error between the reconstructed features and the original features corresponding to the selected keyframes, such that:
+
+$$
+\mathcal { L } _ { r } = \frac { 1 } { X } \sum _ { i \in X } | | \mathbf { x } _ { i } - \hat { \mathbf { y } } _ { i } | | _ { 2 } ,
+$$
+
+where $\hat { \mathbf { y } }$ denotes the reconstructed features.
+
+Diversity Loss. We employ a repelling regularizer [45] to enforce diversity among selected keyframes. Similar to [31, 30], we compute the diversity loss, $\mathcal { L } _ { d }$ , as the pairwise cosine similarity between the selected keyframes:
+
+$$
+\mathcal { L } _ { d } = \frac { 1 } { X ( X - 1 ) } \sum _ { i \in X } \sum _ { j \in X , j \neq i } \frac { \hat { \mathbf { y } } _ { i } \cdot \hat { \mathbf { y } } _ { j } } { \vert \vert \hat { \mathbf { y } } _ { i } \vert \vert _ { 2 } \cdot \vert \vert \hat { \mathbf { y } } _ { j } \vert \vert _ { 2 } } ,
+$$
+
+where $\hat { \mathbf { y } } _ { i }$ and $\hat { \mathbf { y } _ { j } }$ denote the reconstructed feature vectors of the $i$ -th and $j$ -th node.
+
+The final loss function for supervised learning is then,
+
+$$
+\mathcal { L } _ { s u p } = \alpha \cdot \mathcal { L } _ { c } + \beta \cdot \mathcal { L } _ { d } + \lambda \cdot \mathcal { L } _ { r } ,
+$$
+
+where $\alpha , \beta ,$ , and $\lambda$ control the trade-off between the three loss functions. We modify the loss function to extend CLIP-It to the unsupervised video summarization setting. We omit $\mathcal { L } _ { c }$ since the groundtruth summary cannot be used for supervision and represent the final loss function for unsupervised learning as:
+
+$$
+\mathcal { L } _ { u n s u p } = \beta \cdot \mathcal { L } _ { d } + \lambda \cdot \mathcal { L } _ { r } ,
+$$
+
+where $\beta$ and $\lambda$ are balancing parameters to control the trade-off between the two terms. We include implementation details of our method in the Supp.
+
+Table 1: Supervised. Comparing F1 Scores of our methods with supervised baselines on the SumMe [7] and TVSum [36] datasets using Standard, Augment, and Transfer data configurations.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">SumMe</td><td colspan="3">TVSum</td></tr><tr><td>Standard AugmentTransfer Standard AugmentTransfer</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Zhang et al. (SumTransfer) [42]</td><td>40.9</td><td>41.3</td><td>38.5</td><td>-</td><td>=</td><td>1</td></tr><tr><td>Zhang et al. (LSTM) [43]</td><td>38.6</td><td>42.9</td><td>41.8</td><td>54.7</td><td>59.6</td><td>58.7</td></tr><tr><td>Mahasseni et al. (SUM-GANsup) [22]</td><td>41.7</td><td>43.6</td><td>-</td><td>56.3</td><td>61.2</td><td>=</td></tr><tr><td>Rochan et al. (SUM-FCN) [31]</td><td>47.5</td><td>51.1</td><td>44.1</td><td>56.8</td><td>59.2</td><td>58.2</td></tr><tr><td>Rochan et al. (SUM-DeepLab) [31]</td><td>48.8</td><td>50.2</td><td>45.0</td><td>58.4</td><td>59.1</td><td>57.4</td></tr><tr><td>Zhou et al. [47]</td><td>42.1</td><td>43.9</td><td>42.6</td><td>58.1</td><td>59.8</td><td>58.9</td></tr><tr><td> Zhang et al. [44]</td><td>=</td><td>44.9</td><td>1</td><td>=</td><td>63.9</td><td>1</td></tr><tr><td>Fajtl et al. [3]</td><td>49.7</td><td>51.1</td><td>-</td><td>61.4</td><td>62.4</td><td>1</td></tr><tr><td>Rochan et al. [30]</td><td>1</td><td>48.0</td><td>41.6</td><td></td><td>56.1</td><td>55.7</td></tr><tr><td>Chen et al. (V2TS) [1]</td><td>-</td><td>1</td><td>1</td><td>62.1</td><td>1</td><td>1</td></tr><tr><td>He et al. [9]</td><td>47.2</td><td>-</td><td>-</td><td>59.4</td><td>-</td><td>-</td></tr><tr><td>Park et al. (SumGraph) [26]</td><td>51.4</td><td>52.9</td><td>48.7</td><td>63.9</td><td>65.8</td><td>60.5</td></tr><tr><td>GoogleNet+bi-LSTM</td><td>38.5</td><td>42.4</td><td>40.7</td><td>53.9</td><td>59.6</td><td>58.6</td></tr><tr><td>ResNet+bi-LSTM</td><td>39.4</td><td>44.0</td><td>42.6</td><td>55.0</td><td>61.0</td><td>59.9</td></tr><tr><td>CLIP-Image+bi-LSTM</td><td>41.1</td><td>45.9</td><td>44.9</td><td>56.8</td><td>63.7</td><td>61.6</td></tr><tr><td>CLIP-Image+Video Caption+bi-LSTM</td><td>41.2</td><td>46.1</td><td>45.5</td><td>57.1</td><td>64.3</td><td>62.4</td></tr><tr><td>GoogleNet+Transformer</td><td>51.6</td><td>53.5</td><td>49.4</td><td>64.2</td><td>66.3</td><td>61.3</td></tr><tr><td>ResNet+Transformer</td><td>52.8</td><td>54.9</td><td>50.3</td><td>65.0</td><td>67.5</td><td>62.8</td></tr><tr><td>CLIP-Image+Transformer</td><td>53.5</td><td>55.3</td><td>51.0</td><td>65.5</td><td>68.1</td><td>63.4</td></tr><tr><td>CLIP-It: CLIP-Image+Video Caption+Transformer</td><td>54.2</td><td>56.4</td><td>51.9</td><td>66.3</td><td>69.0</td><td>65.5</td></tr></table>
+
+# 4 Experiments
+
+In this section, we describe the experimental setup and evaluation of our method on two tasks: generic video summarization and query-focused video summarization.
+
+# 4.1 Generic Video Summarization
+
+Generic video summarization involves generating a single general-purpose summary to describe the input video. Note that while prior works only use visual cues from the video, our method also allows for video captions as an input feature. For a fair comparison, we include ablations of our method that do not use any language cues and only the visual features used in earlier works [43].
+
+Datasets. We evaluate our approach on two standard video summarization datasets (TVSum [36] and SumMe [7]) and on the generic summaries for UT Egocentric videos [16] provided by the QFVS dataset [33]. TVSum [36] consists of 50 videos pertaining to 10 categories (how to videos, news, documentary, etc) with 5 videos from each category, typically 1-5 minutes in length. SumMe [7] consists of 25 videos capturing multiple events such as cooking and sports, and the lengths of the videos vary from 1 to 6 minutes. In addition to training on each dataset independently, we follow prior work and augment training data with 39 videos from the YouTube dataset [2] and 50 videos from the Open Video Project (OVP) dataset [24]. YouTube dataset consists of news, sports and cartoon videos. OVP dataset consists of multiple different genres including documentary videos. These datasets are diverse in nature and come with different types of annotations, frame-level scores for TVSum and shot-level scores for SumMe. They are integrated to create the ground-truth using the procedure in [43]. The UT Egocentric dataset consists of 4 videos captured from head-mounted cameras. Each video is about 3-5 hours long, captured in a natural, uncontrolled setting and contains a diverse set of events. The QFVS dataset [33] provides ground-truth generic summaries for these 4 videos. The summaries were constructed by dividing the video into shots and asking 3 users to select the relevant shots. The final ground-truth is an average of annotations from all users.
+
+Note. All the datasets - YouTube [2], Open Video Project (OVP) dataset [24], TVSum [36], SumMe [7], and QFVS [32] were collected by the creators (cited) and consent for any personally identifiable information (PII) was ascertained by the authors where necessary.
+
+Data configuration for TVSum and SumMe. Following previous works [43, 42, 31, 26], we evaluate our approach in three different data settings: Standard, Augment, and Transfer. In the Standard setting, the training and test splits are from the same dataset (i.e. either TVSum or SumMe). For SumMe we use available splits, and for TVSum we randomly select $20 \%$ of the videos for testing and construct 5 different splits and report an average result on all 5 splits. For the Augment setting, the training set from one dataset (e.g., TVSum) is combined with all the data from the remaining three datasets (e.g., SumMe, OVP, and YouTube). This setting yields the best performing models due to the additional training data. The Transfer setting is the most challenging of the three. It involves training a model on three datasets and evaluating on the fourth unseen dataset.
+
+Table 2: Unsupervised. Comparing F1 Scores of our methods with unsupervised baselines on the SumMe [7] and TVSum [36] datasets using Standard, Augment, and Transfer data configurations.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">SumMe</td><td colspan="3">TVSum</td></tr><tr><td colspan="6">Standard AugmentTransfer Standard AugmentTransfer</td></tr><tr><td>Mahasseni et al. [22]</td><td>39.1</td><td>43.4</td><td></td><td>51.7</td><td>59.5</td><td></td></tr><tr><td>Yuan et al. [41]</td><td>41.9</td><td>1</td><td>=</td><td>57.6</td><td>1</td><td></td></tr><tr><td>Rochan et al. (SUM-FCNun sup) [31]</td><td>41.5</td><td>1</td><td>39.5</td><td>52.7</td><td>1</td><td>-</td></tr><tr><td>Rochan et al. [30]</td><td>47.5</td><td>=</td><td>41.6</td><td>55.6</td><td>=</td><td>55.7</td></tr><tr><td>He et al. [9]</td><td>46.0</td><td>47.0</td><td>44.5</td><td>58.5</td><td>58.9</td><td>57.8</td></tr><tr><td>Park et al. (SumGraph) [26]</td><td>49.8</td><td>52.1</td><td>47.0</td><td>59.3</td><td>61.2</td><td>57.6</td></tr><tr><td>GoogleNet+bi-LSTM</td><td>33.1</td><td>38.0</td><td>36.5</td><td>47.7</td><td>54.9</td><td>52.3</td></tr><tr><td>ResNet+bi-LSTM</td><td>34.5</td><td>40.1</td><td>39.6</td><td>51.0</td><td>56.2</td><td>53.8</td></tr><tr><td>CLIP-Image+bi-LSTM</td><td>35.7</td><td>41.0</td><td>41.4</td><td>52.8</td><td>58.7</td><td>56.0</td></tr><tr><td>CLIP-Image+Video Caption+bi-LSTM</td><td>36.9</td><td>42.4</td><td>42.5</td><td>53.5</td><td>59.4</td><td>57.6</td></tr><tr><td>GoogleNet+Transformer</td><td>50.0</td><td>52.7</td><td>47.6</td><td>59.9</td><td>62.1</td><td>58.4</td></tr><tr><td>ResNet+Transformer</td><td>50.8</td><td>53.9</td><td>49.3</td><td>61.1</td><td>63.0</td><td>59.9</td></tr><tr><td>CLIP-Image+Transformer</td><td>51.2</td><td>53.6</td><td>49.2</td><td>61.9</td><td>64.0</td><td>60.6</td></tr><tr><td>CLIP-It: CLIP-Image+Video Caption+Transformer</td><td>52.5</td><td>54.7</td><td>50.0</td><td>63.0</td><td>65.7</td><td>62.8</td></tr></table>
+
+Table 3: Comparing F1 Scores for generic video summarization on the QFVS datase   
+
+<table><tr><td>Supervised</td><td>Vid 1</td><td>Vid 2</td><td>Vid 3</td><td>Vid 4</td><td>Avg</td></tr><tr><td>SubMod [6]</td><td>49.51</td><td>51.03</td><td>64.52</td><td>35.82</td><td>50.22</td></tr><tr><td>QFVS [33]</td><td>62.66</td><td>46.11</td><td>58.85</td><td>33.50</td><td>50.29</td></tr><tr><td>CLIP-Image + bi-LSTM ResNet +Transformer</td><td>65.43</td><td>56.55</td><td>68.63</td><td>40.06</td><td>57.67</td></tr><tr><td></td><td>66.97</td><td>58.32</td><td>70.10</td><td>43.31</td><td>59.67</td></tr><tr><td>CLIP-Image + Transformer</td><td>70.8</td><td>61.67</td><td>72.43</td><td>47.48</td><td>63.11</td></tr><tr><td>CLIP-It (Gen. Caption)</td><td>74.13</td><td>63.44</td><td>75.86</td><td>50.23</td><td>65.92</td></tr><tr><td>CLIP-It (GT Caption)</td><td>84.98</td><td>71.26</td><td>82.55</td><td>61.46</td><td>75.06</td></tr><tr><td> Unsupervised</td><td>Vid 1</td><td>Vid 2</td><td>Vid 3</td><td>Vid 4</td><td>Avg</td></tr><tr><td>Quasi [46]</td><td>53.06</td><td>53.80</td><td>49.91</td><td>22.31</td><td>44.77</td></tr><tr><td>CLIP-Image + Transformer</td><td>65.44</td><td>57.21</td><td>65.10</td><td>41.63</td><td>57.35</td></tr><tr><td>CLIP-It (Gen. Caption)</td><td>67.02</td><td>59.48</td><td>66.43</td><td>44.19</td><td>59.28</td></tr><tr><td>CLIP-It (GT Caption)</td><td>73.90</td><td>66.83</td><td>75.44</td><td>52.31</td><td>67.12</td></tr></table>
+
+Quantitative Results. We compare our method and its ablations to supervised video summarization baselines in Tab.1. We report F1 scores on the TVSum and SumMe datasets for all three data settings. Our full method, CLIP-It (CLIP-Image $+$ Video Caption $^ +$ Transformer), described in Sec. 3, outperforms state-of-the-art by a large margin on all three settings. Particularly, in the Transfer setting we outperform the previous state-of-the-art SumGraph [26] by $5 \%$ on TVSum and $3 \%$ on SumMe, indicating that our model is better than the baselines in generalizing to out-of-distribution data.
+
+To prove the effectiveness of each component of our model, we include comparisons to different ablations. In CLIP-Image+Transformer, we ablate the Language-Guided Attention module and directly pass the CLIP-Image features as input to the transformer. As seen, the performance drops by $2 \%$ indicating that conditioning on CLIP language embeddings leads to better summaries. Substituting the Frame-Scoring Transformer with a Bidirectional LSTM in CLIP-Image+bi-LSTM and CLIP-Image $^ +$ Video Caption+bi-LSTM again results in a performance drop, thus highlighting the need for the Transformer module in our model. For a fair comparison with the baselines, in GoogleNet+Transformer we use the same GoogleNet features provided by Zhang et al. [43] and used by all the other baselines and do not include language features. This method still outperforms Sum
+
+![](images/7358c9804a60b78676570b2d26780d98322d1dee29fa09714b106bd270f41b48.jpg)  
+Figure 3: Comparison of ground-truth summary to results from CLIP-Image $^ +$ Transformer and the full CLIP-It model (CLIP-Image $^ +$ Video Caption $^ +$ Transformer). The input is a recipe video. Without captions, the model assigns high scores to certain irrelevant frames such as scenes of the woman talking or eating which hurts the precision. With captions, the cross-attention mechanism ensures that frames with important actions and objects are assigned high scores.
+
+![](images/3c1b4747417c0ee5ce5f793081633a4249bd4d0f0c0be562dedda1199fe5b01d.jpg)  
+Figure 4: Qualitative result comparing the generic summary from CLIP-It with the ground-truth summary. The plots showing predicted and ground-truth frame-level scores are similar, indicating that frames that were given a high score in ground-truth were also assigned high scores by our model.
+
+Graph [26]. Replacing the GoogleNet features with ResNet features results in a small performance improvement but CLIP-Image features prove to be most effective.
+
+Tab. 2 compares F1 scores in the unsupervised setting to other unsupervised baselines. CLIP-It (CLIP-Image $+$ Video Caption+Transformer), outperforms the best performing baseline on all settings on both datasets. We again observe large performance improvements in the Transfer setting and notice that overall our unsupervised method performs almost as well as our supervised counterpart. In CLIP-Image+Transformer we ablate the language component which causes a drop in performance, thus proving that language cues are useful in the unsupervised setting as well. Other ablations yield similar results reiterating the need for each component of our method. We also follow Otani et al. [25] and report results on rank based metrics, Kendall’s $\tau$ [14] and Spearman’s $\rho$ [49] correlation coefficients in the Supp.
+
+Table 4: Results on UT Egocentric dataset [16]   
+
+<table><tr><td>Method</td><td>F-Measure</td><td>Recall</td></tr><tr><td colspan="3">Supervised</td></tr><tr><td>Submod-V+Both et al. [27]</td><td>34.15</td><td>31.59</td></tr><tr><td>CLIP Image + Transformer</td><td>41.58</td><td>39.96</td></tr><tr><td>CLIP-It: CLIP Image + Gen. Video Caption + Transformer</td><td>44.70</td><td>43.28</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption + Transformer</td><td>52.10</td><td>50.76</td></tr><tr><td colspan="3">Unsupervised</td></tr><tr><td>CLIP Image + Transformer</td><td>39.22</td><td>37.46</td></tr><tr><td>CLIP-It: CLIP Image + Gen.Video Caption + Transformer</td><td>42.10</td><td>40.65</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption + Transformer</td><td>49.98</td><td>47.91</td></tr><tr><td colspan="3">Table 5: Results on TV Episodes dataset [16]</td></tr><tr><td>Method</td><td>F-Measure</td><td>Recall</td></tr><tr><td colspan="3">Supervised</td></tr><tr><td>Submod-V+Sem. Rep. et al. [27]</td><td>40.90</td><td>37.02</td></tr><tr><td>CLIP Image+ Transformer</td><td>47.82</td><td>46.02</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption + Transformer</td><td>55.34</td><td>53.90</td></tr><tr><td colspan="3">Unsupervised</td></tr><tr><td>CLIP Image + Transformer</td><td>45.77</td><td>44.01</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption+ Transformer</td><td>53.42</td><td>52.50</td></tr></table>
+
+Tab. 3 shows F1 scores for the generic setting on the QFVS Dataset [33]. Following [33], we run four rounds of experiments leaving out one video for testing and one for validation, while keeping the remaining two for training. Our method, CLIP-It (Gen. Captions) outperforms both supervised and unsupervised baselines by a large margin on all four videos, and particularly on Video 4, which happens to be the most difficult for all methods. Adding captions helps significantly improve $( 2 \% )$ the summaries and outperforms the CLIP Image $^ +$ Transformer baseline. To see how well our model would perform if we had perfect captions, we also show results by using the ground-truth captions obtained from VideoSet [40]. Replacing CLIP-Image features with ResNet features causes a drop in performance. Likewise, replacing the Transformer with a bi-LSTM also hurts performance.
+
+We include results of our method (1) without captions (2) using generated captions (3) using the ground truth captions provided by VideoSet [40] for UT Egocentric and TV Episodes datasets in Tables 4 and 5. As ground truth, we obtain 15 summaries for each video using the same greedy n-gram matching and ordered subshot selection procedures as previous work [27]. We follow the same procedure as in prior work [40, 27] for creating and evaluating text summaries from video summaries. Our method outperforms [27] in both the supervised and unsupervised settings on both datasets.
+
+Qualitative Results. We highlight the need to use language, specifically dense video captions, for constructing generic video summaries through a qualitative example in Fig. 3. The input is a video of a woman demonstrating how to make a chicken sandwich. The ground truth summary shows the scores computed by averaging the annotations from all users as in [43] and the keyframes that received high scores. Next we show the result from the baseline CLIP-Image+Transformer, which uses only visual features and no language input. The predicted scores show that the high scoring frames in the ground truth also receive a high score by our baseline, however a lot of irrelevant frames end up receiving a high score too. Thus, when the model finally picks the keyframes it ends up selecting frames where the person is talking or eating the sandwich (shown in red) which do not correspond to the key steps in the video. Adding language guidance via generated captions helps address this problem. The last row shows the captions generated by BMT [11]. The results shown are from our full CLIP-It model. The predicted scores are more similar to ground truth scores and the highest scoring keyframes are the same as the ground truth.
+
+Table 6: Comparing F1 Scores of different methods on the QFVS dataset.   
+
+<table><tr><td></td><td>Vid 1</td><td>Vid 2</td><td>Vid 3</td><td>Vid 4</td><td>Avg</td></tr><tr><td>SeqDPP[51</td><td>36.59</td><td>43.67</td><td>25.26</td><td>18.15</td><td>30.92</td></tr><tr><td>SH-DPP [32]</td><td>35.67</td><td>42.74</td><td>36.51</td><td>18.62</td><td>33.38</td></tr><tr><td>QFVS [33]</td><td>48.68</td><td>41.66</td><td>56.47</td><td>29.96</td><td>44.19</td></tr><tr><td>CLIP-Image + Query + bi-LSTM</td><td>54.47</td><td>48.59</td><td>62.81</td><td>38.64</td><td>51.13</td></tr><tr><td>ResNet + Query + Transformer</td><td>55.19</td><td>51.03</td><td>64.26</td><td>39.47</td><td>52.49</td></tr><tr><td>CLIP-It: CLIP-Image + Query + Transformer</td><td>57.13</td><td>53.60</td><td>66.08</td><td>41.41</td><td>54.55</td></tr></table>
+
+![](images/8598abec91315572d62925e9e2ddce1b0bafa26350dcb4469546767f520beafd.jpg)  
+Figure 5: Result of our method on the QFVS dataset. The first row shows some frames from the 4 hour long input video. Given the query "book and chair", Summary 1 shows some frames selected by our method. Summary 2 shows frames for the query "Sun and Tree".
+
+Fig. 4 shows compares summaries generated by our method to the ground truth summary on a cooking video from the SumMe dataset. Predicted scores are the frame-wise scores predicted by CLIP-It and GT Scores are the scores computed from user annotations [43]. The top row (True Positives) shows high scoring keyframes which are chosen by both our method and the ground truth, and the green arrows point to the assigned scores. As we see, they are clear, distinct and represent key actions in the recipe. The bottom row (True Negatives) shows frames which are assigned a low score (shown by the red arrows) and are not part of the final summaries (both GT and predicted). E.g., the first frame is irrelevant and corresponds to a segment between key steps, while the second frame has poor lighting and its hard to tell what is being done.
+
+# 4.2 Query-Focused Video Summarization
+
+In Query-Focused Video Summarization, the summarization process is conditioned on an input user query, thus, multiple summaries can be obtained for the same video using different input queries.
+
+Dataset. We evaluate our method on the QFVS dataset based on the UT Egocentric videos described earlier. The dataset consists of 46 queries for each of the four videos and user-annotated video summaries for each query.
+
+Quantitative Results. In Tab. 6, we compare F1 scores of our method to 3 baselines, SH-DPP [32], Seq-DPP [5], and QFVS [33]. Following [33], we run four rounds of experiments leaving out one video for testing and one for validation, while keeping the remaining two for training. Our full model achieves an avg F1 score of $5 4 . 5 5 \%$ outperforming the best baseline $( 4 4 . 1 9 \% )$ by $10 \%$ . We would like to point out that our method uses more recent image features compared to the baselines. At the same time, when switching from CLIP to ResNet image embedding, we still improve significantly over the baselines. We expect the improvement to also hold with weaker features (e.g., as demonstrated with GoogleNet results in Tab. 1, 2).
+
+Qualitative Results. Fig. 5 shows a result on the QFVS dataset for UT Egocentric videos. The input is an egocentric video shot from a head-mounted camera and spans 4 hours. It consists of multiple events from a person’s day, such as reading books, working on the laptop, walking in the streets, and so on. Summary 1 and 2 show results from our CLIP-It method when the input query is “Book and chair” and “Sun and tree” respectively. The frames shown are frames assigned high-scores by our method. As seen, given the same input video, different queries yield different summaries.
+
+# 5 Discussion and Broader Impacts
+
+We introduced CLIP-It, a unified language-guided framework for generic and query focused video summarization. Video summarization is a relevant problem with many use-cases, and our approach provides greater flexibility to the users, allowing them to guide summarization with open-ended natural language queries. We envision potential positive impact from improved user experience if adopted on video platforms such as YouTube. We rely on an off-the-shelf video captioning model [11] and a large-scale vision-language model (CLIP [29]) which may have encoded some inappropriate biases that could propagate to our model. Our visual inspection of the obtained summarization results did not raise any apparent concerns. However, practitioners who wish to use our approach should be mindful of the sources of bias we have outlined above depending on the specific use case they are addressing.
+
+Acknowledgements. We thank Arun Mallya for very helpful discussions and feedback. We’d also like to thank Huijuan Xu for feedback on the draft. This work was supported in part by DoD including DARPA’s XAI, LwLL, and SemaFor programs, as well as BAIR’s industrial alliance programs.
+
+# References
+
+[1] Chen, B.C., Chen, Y.Y., Chen, F.: Video to text summary: Joint video summarization and captioning with recurrent neural networks. British Machine Vision Conference (BMVC) (2017) 3, 6 [2] De Avila, S.E.F., Lopes, A.P.B., da Luz Jr, A., de Albuquerque Araújo, A.: Vsumm: A mechanism designed to produce static video summaries and a novel evaluation method. Patt. Rec. Letters (2011) 6   
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+        "text": "CLIP-It! Language-Guided Video Summarization ",
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+        "text": "Medhini Narasimhan Anna Rohrbach Trevor Darrell ",
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+        "text": "University of California, Berkeley {medhini, anna.rohrbach, trevordarrell}@berkeley.edu https://medhini.github.io/clip_it ",
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+            "Figure 1: We introduce CLIP-It, a language-guided multimodal transformer for generic and query-focused video summarization. The figure shows results from our method. Given a day-long video of a national park tour, the generic summary (top) is a video with relevant and diverse keyframes. When using the query “All the scenes containing restaurants and shopping centers”, the generated query-focused summary includes all the matching scenes. Similarly, the query “All water bodies such as lakes, rivers, and waterfalls”, yields a short summary containing all the water bodies present in the video. "
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+        "text": "Abstract ",
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+        "text": "A generic video summary is an abridged version of a video that conveys the whole story and features the most important scenes. Yet the importance of scenes in a video is often subjective, and users should have the option of customizing the summary by using natural language to specify what is important to them. Further, existing models for fully automatic generic summarization have not exploited available language models, which can serve as an effective prior for saliency. This work introduces CLIP-It, a single framework for addressing both generic and queryfocused video summarization, typically approached separately in the literature. We propose a language-guided multimodal transformer that learns to score frames in a video based on their importance relative to one another and their correlation with a user-defined query (for query-focused summarization) or an automatically generated dense video caption (for generic video summarization). Our model can be extended to the unsupervised setting by training without ground-truth supervision. We outperform baselines and prior work by a significant margin on both standard video summarization datasets (TVSum and SumMe) and a query-focused video summarization dataset (QFVS). Particularly, we achieve large improvements in the transfer setting, attesting to our method’s strong generalization capabilities. ",
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+        "text": "1 Introduction ",
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+        "text": "An effective video summary captures the essence of the video and provides a quick overview as an alternative to viewing the whole video; it should be succinct yet representative of the entire video. Summarizing videos has many use cases - for example, viewers on YouTube may want to watch a short summary of the video to assess its relevance. While a generic summary is useful for capturing the important scenes in a video, it is even more practical if the summary can be customized by the user. As seen in Fig. 1, users should be able to indicate the concepts of the video they would like to see in the summary using natural language queries. ",
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+        "type": "text",
+        "text": "Generic video summarization datasets such as TVSum [36] and SumMe [7] provide ground-truth annotations in the form of frame or shot-level importance scores specified by multiple annotators. Several learning-based methods reduce the task to a frame-wise score prediction problem. Sequence labeling methods [5, 42, 43, 44, 18] model variable-range dependencies between frames but fail to capture relationships across all frames simultaneously. While attention [3] and graph based [26] methods address this partly, they disregard ordering of the input frames which is useful when predicting scores. Moreover, these methods use only visual cues to produce the summaries and cannot be customized with a natural language input. Another line of work, query-focused video summarization [32], allows the users to customize the summary by specifying a query containing two concepts (eg., food and drinks). However, in their work the query can only be chosen from a fixed set of predefined concepts which limits the flexibility for the user. ",
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+        "text": "It is important to note that efforts in generic and query-focused video summarization have so far been disjoint, with no single method for both. Our key innovation is to unify these tasks in one language-guided framework. We introduce CLIP-It, a multimodal summarization model which takes two inputs, a video and a natural language text, and synthesizes a summary video conditioned on the text. In case of generic video summarization, the natural language text is a video description obtained using an off-the-shelf dense video captioning method. Alternatively, in the case of query-focused video summarization, the language input is a user-defined query. Unlike existing generic methods which only use visual cues, we show that adding language as an input leads to the “discovery” of relevant concepts and actions resulting in better summaries. Our method uses a Transformer with positional encoding, which can attend to all frames at once (unlike LSTM based methods) and also keep track of the ordering of frames (unlike graph based methods). In contrast to existing query-focused methods [32] which only allow keyword based queries, we aim to enable open-ended natural language queries for users to customize video summaries. For example, as seen in Fig. 1, using our method users can specify long and descriptive queries such as, “All water bodies such as lakes, rivers, and waterfalls” which is not possible with previous methods. ",
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+    {
+        "type": "text",
+        "text": "Given an input video, CLIP-It generates a video summary guided by either a user-defined natural language query or a system generated description. It uses a Language-Guided Attention head to compute a joint representation of the image and language embeddings, and a Frame-Scoring Transformer to assign scores to individual frames in the video using the fused representations. Following [43, 42], the summary video is constructed from high scoring frames by converting framelevel scores to shot-level scores and using knapsack algorithm to fit the maximum number of high scoring shots in a timed window. Fig. 1 shows an output from our method. Given an hour long video of a national park tour, we generate a 2 minute generic summary comprising of all the important scenes in the video. Given the two language queries, our method picks the matching keyframes in both cases. We can train CLIP-It without ground-truth supervision by leveraging the reconstruction and diversity losses [9, 31]. For generic video summarization, we evaluate our approach on the standard benchmarks, TVSum and SumMe. We achieve performance improvement on F1 score of nearly $3 \\%$ in the supervised setting and $4 \\%$ in the unsupervised setting on both datasets. We show large gains $( 5 \\% )$ in the Transfer setting, where CLIP-It is trained and evaluated on disjoint sets of data. For the queryfocused scenario we evaluate on the QFVS dataset [33], where we also achieve state-of-the-art results. ",
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+        "type": "text",
+        "text": "To summarize our contributions, we introduce CLIP-It, a language-guided model that unifies generic and query-focused video summarization. Our approach uses language conditioning in the form of off-the-shelf video descriptions (for generic summarization) or user-defined natural language queries (for query-focused summarization). We show that the Transformer design enables effective contextualization across frames, benefiting our tasks. We also demonstrate the impact of language guidance on generic summarization. Finally, we establish the new state-of-the-art on both generic and query-focused datasets in supervised and unsupervised settings. ",
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+        "text": "2 Related Work ",
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+        "text": "Generic Video Summarization. A video summary is a short synopsis created by stitching together important clips from the original video. Early works [21, 35, 34] referred to it as Video Skimming or Dynamic Video Summarization and used hand-designed features to generate summaries. Likewise, non-parametric unsupervised methods [13, 16, 23, 19, 20, 28] used various heuristics to represent the importance of frames. Introduction of benchmark datasets such as TVSum [36] and SumMe [7] provided relevance scores for frames in videos annotated by users, resulting in multiple human generated summaries. This enabled automatic evaluation of video summarization techniques and has lead to the development of many supervised learning based methods [6, 8, 22, 26, 31, 30, 41, 42, 43, 44, 45]. These approaches capture high-level semantic information and outperform the heuristic unsupervised methods. Fully convolutional sequence networks [31] treat video summarization as a binary label prediction problem. Determinantal point processes [15] and LSTM [10] based approaches [5, 18, 42, 43, 44] model variable-range dependencies between frames. However, these are sequential and fail to capture relationships across all frames simultaneously. Attention [3] and graph based methods [26] address this issue by modeling relationships across all frames, but they disregard the ordering of frames in a video, which is also useful for summarization. Our method uses a Transformer [38] with positional encoding, which allows for joint attention across all frames while maintaining an ordering of the input sequence of frames. ",
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+        "type": "text",
+        "text": "Some of the above works have supervised and unsupervised variants with modifications to the objective function. Specifically, for the unsupervised variant, they use reconstruction and diversity losses which do not require ground truth. We follow prior work in terms of using the same loss functions. Other notable unsupervised approaches include an adversarial LSTM based method [22], a generative adversarial network to learn from unpaired data [30], and a cycle consistent learning objective [41]. ",
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+        "type": "text",
+        "text": "Video-Text Summarization. Plummer et al. [27] use image-language embeddings for generating video summaries but evaluate their approach in the text domain, and not on the generic video summarization benchmarks. Furthermore, they require text to be provided as input and don’t have a mechanism to generate captions if absent. On the other hand, our method works well with off-theshelf captions and we evaluate on both generic and query-focused benchmark datasets. Chen et al. [1] jointly train a text and video summarization network but rely on ground-truth text summaries. ",
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+    {
+        "type": "text",
+        "text": "Query-Focused Video Summarization. Oftentimes users browsing videos on YouTube are looking for something specific so a generic summary might not suffice. In this case, there should be an option to customize the generated summary using a natural language query. Sharghi et al. [32] introduce the Query-Focused Video Summarization (QFVS) dataset for UT Egocentric [16] videos containing user-defined video summaries for a set of pre-defined concepts. Sharghi et al. [33] propose a memory network to attend over different video frames and shots with the user query as input. However, this recurrent attention mechanism precludes parallelization and limits modeling long-range dependencies, which is overcome by our Transformer architecture. Moreover, their method only works with the pre-defined set of keyword based queries in QFVS dataset. Since we use CLIP [29] to encode the language input and train our method on dense video descriptions, this allows users to define freeform queries at test time (as seen in Fig. 1). Other works [12, 39] similarly condition the summary generation on keyword based queries but haven’t released their data. ",
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+        "type": "text",
+        "text": "Transformers. Transformers [38] were introduced for neural machine translation and have since been applied to video-language tasks such as video-retrieval [4] and video captioning [17, 48]. In this work we adapt transformers for video summarization. We modify self-attention [38] to a LanguageGuided Attention block that accepts inputs from two modalities. Additionally, our method also relies on CLIP [29] for extracting image and text features and a Bi-Modal Transformer [11] for dense video caption generation, both of which also have transformer backbone. ",
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+        "type": "text",
+        "text": "3 CLIP-It: Language-Guided Video Summarization ",
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+        "text": "CLIP-It is a unified language-guided framework for both generic and query-focused video summarization. It takes an input video and a user-defined query or a system generated dense video caption and constructs a summary by selecting key frames from the video. First, we explain the intuition behind our approach. In the case of query-focused summarization, clearly, it is necessary to model the user query as input in order to produce an appropriate summary. In the case of generic video summarization no user query is available; nonetheless, we show here that we can leverage the semantic information contained in associated natural language descriptions to guide the summarization process. Assuming we had a generic description that accompanies a video (e.g., A person is walking a dog. The person throws a ball. The dog runs after it.), we could leverage its semantic embedding to match $i t$ to the most relevant portions of the video. Such a description could be obtained automatically by generating dense video captions [11]. We first present an overview of our approach, CLIP-It, followed by a detailed description of the individual components. ",
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+        "image_caption": [
+            "Figure 2: Overview of CLIP-It. Given an input video, CLIP-It generates a summary conditioned on either a user-defined natural language query or an automatically generated dense video caption. The Language-Guided Attention head fuses the image and language embeddings and the Frame-Scoring Transformer jointly attends to all frames to predict their relevance scores. During inference, the video summary is constructed by converting frame scores to shot scores and using Knapsack algorithm to select high scoring shots. "
+        ],
+        "image_footnote": [],
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+        "text": "Overview. Our approach, CLIP-It, is outlined in $\\mathrm { F i g } 2$ . Given a video, we extract $N$ frames denoted by $F _ { i }$ , $i \\in [ 1 , \\dots \\bar { N } ]$ . We formulate the video summarization task as a per-frame binary classification problem. We embed the frames using a pretrained network $f _ { i m g }$ . If a query is provided (in the form of a natural language string), we embed the query using a pretrained network $f _ { t x t }$ . Alternatively, as seen in the figure, we use an off-the-shelf video captioning model to generate a dense video caption with $M$ sentences denoted by $C _ { j }$ , $j \\in [ 1 , \\bar { \\dots { M } } ]$ and embed each sentence using the pretrained network $f _ { t x t }$ . Next, we compute language attended image embeddings $I ^ { * }$ using learned Language-Guided Multi-head Attention $f _ { i m g \\_ t x t } ^ { * }$ . Finally, we train a Frame-Scoring Transformer which assigns scores to each frame in the video (green indicating a high score and red indicating a low score). To construct the video summary during inference, we convert frame-level scores to shot-level scores and finally, use 0/1 knapsack algorithm to pick the key shots [43]. In the following, we describe the Language-Guided Attention and the Frame-Selection Transformer modules, followed by discussing the visual and language encoders. ",
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+    {
+        "type": "text",
+        "text": "Language-Guided Attention. We use Language-Guided Multi-head Attention to efficiently fuse information across the video and language modalities and to infer long-term dependencies across both. Using a single attention head does not suffice as our goal is to allow all captions to attend to all frames in the video. We modify the Multi-Head Attention described in Vaswani et al. [38] to take in inputs from both modalities. We set Query $Q$ , Key $K$ , and Value $V$ as follows, ",
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+        "type": "equation",
+        "img_path": "images/cb199205e7555bb1154bf2fb4a6e5141233c44b2977fd77e0a7e8f33b9c60c16.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\therefore , \\nu \\ - \\jmath \\iota x \\iota \\mathrm { ( } \\mathrm { V } _ { \\mathcal { I } } \\mathrm { ) , ~ } \\mathrm { w a n g e - } \\mathrm { G u i d e d ~ A t t n . } ( Q , K , V ) = \\mathrm { C o n c a t ( h e a d _ 1 , } . . . , \\mathrm { h e a d _ h } ) W ^ { O } , } } \\\\ { { \\mathrm { L a n g u a g e - } \\mathrm { G u i d e d ~ A t t n . } ( Q , K , V ) = \\mathrm { C o n c a t ( h e a d _ 1 , } . . . , \\mathrm { h e a d _ h } ) W ^ { O } , } } \\\\ { { \\mathrm { w h e r e ~ h e a d _ i = A t t e n t i o n } ( Q W _ { i } ^ { Q } , K W _ { i } ^ { K } , V W _ { i } ^ { V } ) } } \\\\ { { \\mathrm { a n d ~ A t t e n t i o n } ( Q , K , V ) = \\mathrm { s o f m a x } ( \\displaystyle \\frac { Q K ^ { T } } { \\sqrt { d _ { k } } } ) V } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\it { W _ { i } ^ { Q } }$ , $W _ { i } ^ { K }$ , and $W _ { i } ^ { V }$ are learned parameter matrices and $d _ { k }$ is the dimensions of $K$ . The output of the Language-Guided Multi-Head Attention are the attended image embeddings, denoted as $F _ { i } ^ { ' }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Frame-Scoring Transformer. Finally, it is also important to ensure that we do not include redundant information, e.g., several key shots from the same event. To better model interactions across frames and contextualize them w.r.t. each other, we add a Frame-Scoring Transformer that takes image-text representations as input and outputs one score per frame. Based on the Transformer model [38], this module assigns relevance scores to the attended image embeddings $F _ { i } ^ { ' }$ . We feed $F _ { i } ^ { ' }$ to the bottom of both the encoder and decoder stacks. Similar to [38], we use positional encoding to insert information about the relative positions of the tokens in the sequence. We add positional encodings to the input embeddings at the bottom of the encoder and decoder stacks. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Image Encoding. We encode the image using a pre-trained network $f _ { i m g }$ . We experiment with the following networks: GoogleNet [37] (for a fair comparison to prior work), ResNet [8], and CLIP [29]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Text Encoding. We encode the user-defined query or the system generated dense caption using a pre-trained network $f _ { t x t }$ . In this case we tried the CLIP (ViT and RN101) model. In case of generic video summarization, we generate dense video captions for the whole video in order to condition on language and incorporate added semantics into our video summarization pipeline. We use the Bi-Modal Transformer [11] dense video captioning model that generates a multi-sentence description given a video with audio. Since there are multiple sentences in the dense video caption, we first embed each sentence of the caption using the text encoder $f _ { t x t }$ described above. They are then concatenated and fused using a multi-layer perceptron (MLP). ",
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+    {
+        "type": "text",
+        "text": "3.1 Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We employ 3 loss functions (classification, diversity, and reconstruction) to train our model. The supervised setting uses all 3 and the unsupervised setting uses only diversity and reconstruction losses. ",
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+        "type": "text",
+        "text": "Classification loss. We use a weighted binary cross entropy loss for classifying each frame: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/97e15b9ce3a49ab5a72cb5809875e3e18ddca3b297d27441b6b535f6d5eafee2.jpg",
+        "text": "$$\n\\mathcal { L } _ { c } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } w ^ { * } [ x _ { i } ^ { * } \\mathrm { l o g } ( x _ { i } ) ] + ( 1 - w ^ { * } ) [ ( 1 - x _ { i } ^ { * } ) \\mathrm { l o g } ( 1 - x _ { i } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { x } _ { i } ^ { * }$ is the ground-truth label of the $i$ -th frame and $N$ is the total number of frames in the $w ^ { * }$ weif ht assigned to the class is a background frame $\\boldsymbol { x } _ { i } ^ { * }$ , which is set to $\\frac { \\# k e y f r a m e s } { N }$ if $\\boldsymbol { x } _ { i } ^ { * }$ is a keyframe and $1 - { \\frac { \\# k e y f r a m e s } { N } }$ $\\boldsymbol { x } _ { i } ^ { * }$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the purpose of training without any supervision, we employ two additional losses that enforce diversity in the selected keyframes. We first select keyframes $X$ based on the scores assigned by the Frame-Scoring Transformer. We pass the output features of the transformer model for the selected keyframes through a decoder network consisting of $1 \\times 1$ convolution layers to obtain reconstructed feature vectors for the selected keyframes such that each keyframe feature vector is of the same dimension as its corresponding input frame-level feature vector. ",
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+    },
+    {
+        "type": "text",
+        "text": "Reconstruction Loss. The reconstruction loss $\\mathcal { L } _ { r }$ is defined as the mean squared error between the reconstructed features and the original features corresponding to the selected keyframes, such that: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cfa3e99f4ed6b7899a92854fede6a172f1b755f8d7055df221b74adb4bf92716.jpg",
+        "text": "$$\n\\mathcal { L } _ { r } = \\frac { 1 } { X } \\sum _ { i \\in X } | | \\mathbf { x } _ { i } - \\hat { \\mathbf { y } } _ { i } | | _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\mathbf { y } }$ denotes the reconstructed features. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Diversity Loss. We employ a repelling regularizer [45] to enforce diversity among selected keyframes. Similar to [31, 30], we compute the diversity loss, $\\mathcal { L } _ { d }$ , as the pairwise cosine similarity between the selected keyframes: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/910f8740d5eb273a22ec55fb15f42428482be59bf06e80987f7ef31eead07a6f.jpg",
+        "text": "$$\n\\mathcal { L } _ { d } = \\frac { 1 } { X ( X - 1 ) } \\sum _ { i \\in X } \\sum _ { j \\in X , j \\neq i } \\frac { \\hat { \\mathbf { y } } _ { i } \\cdot \\hat { \\mathbf { y } } _ { j } } { \\vert \\vert \\hat { \\mathbf { y } } _ { i } \\vert \\vert _ { 2 } \\cdot \\vert \\vert \\hat { \\mathbf { y } } _ { j } \\vert \\vert _ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\mathbf { y } } _ { i }$ and $\\hat { \\mathbf { y } _ { j } }$ denote the reconstructed feature vectors of the $i$ -th and $j$ -th node. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The final loss function for supervised learning is then, ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d8287bbf9be4046f51dd3f56f06d1358bcf676c0bae40776503fdbc844dfc940.jpg",
+        "text": "$$\n\\mathcal { L } _ { s u p } = \\alpha \\cdot \\mathcal { L } _ { c } + \\beta \\cdot \\mathcal { L } _ { d } + \\lambda \\cdot \\mathcal { L } _ { r } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha , \\beta ,$ , and $\\lambda$ control the trade-off between the three loss functions. We modify the loss function to extend CLIP-It to the unsupervised video summarization setting. We omit $\\mathcal { L } _ { c }$ since the groundtruth summary cannot be used for supervision and represent the final loss function for unsupervised learning as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/87ecf7bd16141466368284bd870634928e447d886071afb86350da2b657bc5a0.jpg",
+        "text": "$$\n\\mathcal { L } _ { u n s u p } = \\beta \\cdot \\mathcal { L } _ { d } + \\lambda \\cdot \\mathcal { L } _ { r } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta$ and $\\lambda$ are balancing parameters to control the trade-off between the two terms. We include implementation details of our method in the Supp. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/0a457e51185718c2ccebd26e980cbf0b4be695ca09bbc09811abbe800366204f.jpg",
+        "table_caption": [
+            "Table 1: Supervised. Comparing F1 Scores of our methods with supervised baselines on the SumMe [7] and TVSum [36] datasets using Standard, Augment, and Transfer data configurations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">SumMe</td><td colspan=\"3\">TVSum</td></tr><tr><td>Standard AugmentTransfer Standard AugmentTransfer</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Zhang et al. (SumTransfer) [42]</td><td>40.9</td><td>41.3</td><td>38.5</td><td>-</td><td>=</td><td>1</td></tr><tr><td>Zhang et al. (LSTM) [43]</td><td>38.6</td><td>42.9</td><td>41.8</td><td>54.7</td><td>59.6</td><td>58.7</td></tr><tr><td>Mahasseni et al. (SUM-GANsup) [22]</td><td>41.7</td><td>43.6</td><td>-</td><td>56.3</td><td>61.2</td><td>=</td></tr><tr><td>Rochan et al. (SUM-FCN) [31]</td><td>47.5</td><td>51.1</td><td>44.1</td><td>56.8</td><td>59.2</td><td>58.2</td></tr><tr><td>Rochan et al. (SUM-DeepLab) [31]</td><td>48.8</td><td>50.2</td><td>45.0</td><td>58.4</td><td>59.1</td><td>57.4</td></tr><tr><td>Zhou et al. [47]</td><td>42.1</td><td>43.9</td><td>42.6</td><td>58.1</td><td>59.8</td><td>58.9</td></tr><tr><td> Zhang et al. [44]</td><td>=</td><td>44.9</td><td>1</td><td>=</td><td>63.9</td><td>1</td></tr><tr><td>Fajtl et al. [3]</td><td>49.7</td><td>51.1</td><td>-</td><td>61.4</td><td>62.4</td><td>1</td></tr><tr><td>Rochan et al. [30]</td><td>1</td><td>48.0</td><td>41.6</td><td></td><td>56.1</td><td>55.7</td></tr><tr><td>Chen et al. (V2TS) [1]</td><td>-</td><td>1</td><td>1</td><td>62.1</td><td>1</td><td>1</td></tr><tr><td>He et al. [9]</td><td>47.2</td><td>-</td><td>-</td><td>59.4</td><td>-</td><td>-</td></tr><tr><td>Park et al. (SumGraph) [26]</td><td>51.4</td><td>52.9</td><td>48.7</td><td>63.9</td><td>65.8</td><td>60.5</td></tr><tr><td>GoogleNet+bi-LSTM</td><td>38.5</td><td>42.4</td><td>40.7</td><td>53.9</td><td>59.6</td><td>58.6</td></tr><tr><td>ResNet+bi-LSTM</td><td>39.4</td><td>44.0</td><td>42.6</td><td>55.0</td><td>61.0</td><td>59.9</td></tr><tr><td>CLIP-Image+bi-LSTM</td><td>41.1</td><td>45.9</td><td>44.9</td><td>56.8</td><td>63.7</td><td>61.6</td></tr><tr><td>CLIP-Image+Video Caption+bi-LSTM</td><td>41.2</td><td>46.1</td><td>45.5</td><td>57.1</td><td>64.3</td><td>62.4</td></tr><tr><td>GoogleNet+Transformer</td><td>51.6</td><td>53.5</td><td>49.4</td><td>64.2</td><td>66.3</td><td>61.3</td></tr><tr><td>ResNet+Transformer</td><td>52.8</td><td>54.9</td><td>50.3</td><td>65.0</td><td>67.5</td><td>62.8</td></tr><tr><td>CLIP-Image+Transformer</td><td>53.5</td><td>55.3</td><td>51.0</td><td>65.5</td><td>68.1</td><td>63.4</td></tr><tr><td>CLIP-It: CLIP-Image+Video Caption+Transformer</td><td>54.2</td><td>56.4</td><td>51.9</td><td>66.3</td><td>69.0</td><td>65.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we describe the experimental setup and evaluation of our method on two tasks: generic video summarization and query-focused video summarization. ",
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+    {
+        "type": "text",
+        "text": "4.1 Generic Video Summarization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Generic video summarization involves generating a single general-purpose summary to describe the input video. Note that while prior works only use visual cues from the video, our method also allows for video captions as an input feature. For a fair comparison, we include ablations of our method that do not use any language cues and only the visual features used in earlier works [43]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets. We evaluate our approach on two standard video summarization datasets (TVSum [36] and SumMe [7]) and on the generic summaries for UT Egocentric videos [16] provided by the QFVS dataset [33]. TVSum [36] consists of 50 videos pertaining to 10 categories (how to videos, news, documentary, etc) with 5 videos from each category, typically 1-5 minutes in length. SumMe [7] consists of 25 videos capturing multiple events such as cooking and sports, and the lengths of the videos vary from 1 to 6 minutes. In addition to training on each dataset independently, we follow prior work and augment training data with 39 videos from the YouTube dataset [2] and 50 videos from the Open Video Project (OVP) dataset [24]. YouTube dataset consists of news, sports and cartoon videos. OVP dataset consists of multiple different genres including documentary videos. These datasets are diverse in nature and come with different types of annotations, frame-level scores for TVSum and shot-level scores for SumMe. They are integrated to create the ground-truth using the procedure in [43]. The UT Egocentric dataset consists of 4 videos captured from head-mounted cameras. Each video is about 3-5 hours long, captured in a natural, uncontrolled setting and contains a diverse set of events. The QFVS dataset [33] provides ground-truth generic summaries for these 4 videos. The summaries were constructed by dividing the video into shots and asking 3 users to select the relevant shots. The final ground-truth is an average of annotations from all users. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note. All the datasets - YouTube [2], Open Video Project (OVP) dataset [24], TVSum [36], SumMe [7], and QFVS [32] were collected by the creators (cited) and consent for any personally identifiable information (PII) was ascertained by the authors where necessary. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Data configuration for TVSum and SumMe. Following previous works [43, 42, 31, 26], we evaluate our approach in three different data settings: Standard, Augment, and Transfer. In the Standard setting, the training and test splits are from the same dataset (i.e. either TVSum or SumMe). For SumMe we use available splits, and for TVSum we randomly select $20 \\%$ of the videos for testing and construct 5 different splits and report an average result on all 5 splits. For the Augment setting, the training set from one dataset (e.g., TVSum) is combined with all the data from the remaining three datasets (e.g., SumMe, OVP, and YouTube). This setting yields the best performing models due to the additional training data. The Transfer setting is the most challenging of the three. It involves training a model on three datasets and evaluating on the fourth unseen dataset. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/1e90156464a6e77f236a52b37e7699134f1d634fb362a051e16479788d1b376b.jpg",
+        "table_caption": [
+            "Table 2: Unsupervised. Comparing F1 Scores of our methods with unsupervised baselines on the SumMe [7] and TVSum [36] datasets using Standard, Augment, and Transfer data configurations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">SumMe</td><td colspan=\"3\">TVSum</td></tr><tr><td colspan=\"6\">Standard AugmentTransfer Standard AugmentTransfer</td></tr><tr><td>Mahasseni et al. [22]</td><td>39.1</td><td>43.4</td><td></td><td>51.7</td><td>59.5</td><td></td></tr><tr><td>Yuan et al. [41]</td><td>41.9</td><td>1</td><td>=</td><td>57.6</td><td>1</td><td></td></tr><tr><td>Rochan et al. (SUM-FCNun sup) [31]</td><td>41.5</td><td>1</td><td>39.5</td><td>52.7</td><td>1</td><td>-</td></tr><tr><td>Rochan et al. [30]</td><td>47.5</td><td>=</td><td>41.6</td><td>55.6</td><td>=</td><td>55.7</td></tr><tr><td>He et al. [9]</td><td>46.0</td><td>47.0</td><td>44.5</td><td>58.5</td><td>58.9</td><td>57.8</td></tr><tr><td>Park et al. (SumGraph) [26]</td><td>49.8</td><td>52.1</td><td>47.0</td><td>59.3</td><td>61.2</td><td>57.6</td></tr><tr><td>GoogleNet+bi-LSTM</td><td>33.1</td><td>38.0</td><td>36.5</td><td>47.7</td><td>54.9</td><td>52.3</td></tr><tr><td>ResNet+bi-LSTM</td><td>34.5</td><td>40.1</td><td>39.6</td><td>51.0</td><td>56.2</td><td>53.8</td></tr><tr><td>CLIP-Image+bi-LSTM</td><td>35.7</td><td>41.0</td><td>41.4</td><td>52.8</td><td>58.7</td><td>56.0</td></tr><tr><td>CLIP-Image+Video Caption+bi-LSTM</td><td>36.9</td><td>42.4</td><td>42.5</td><td>53.5</td><td>59.4</td><td>57.6</td></tr><tr><td>GoogleNet+Transformer</td><td>50.0</td><td>52.7</td><td>47.6</td><td>59.9</td><td>62.1</td><td>58.4</td></tr><tr><td>ResNet+Transformer</td><td>50.8</td><td>53.9</td><td>49.3</td><td>61.1</td><td>63.0</td><td>59.9</td></tr><tr><td>CLIP-Image+Transformer</td><td>51.2</td><td>53.6</td><td>49.2</td><td>61.9</td><td>64.0</td><td>60.6</td></tr><tr><td>CLIP-It: CLIP-Image+Video Caption+Transformer</td><td>52.5</td><td>54.7</td><td>50.0</td><td>63.0</td><td>65.7</td><td>62.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/edbbe4b194ce3f4e335a9daf81d51972596e4b6ddfd4062e1716af3d656f231e.jpg",
+        "table_caption": [
+            "Table 3: Comparing F1 Scores for generic video summarization on the QFVS datase "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Supervised</td><td>Vid 1</td><td>Vid 2</td><td>Vid 3</td><td>Vid 4</td><td>Avg</td></tr><tr><td>SubMod [6]</td><td>49.51</td><td>51.03</td><td>64.52</td><td>35.82</td><td>50.22</td></tr><tr><td>QFVS [33]</td><td>62.66</td><td>46.11</td><td>58.85</td><td>33.50</td><td>50.29</td></tr><tr><td>CLIP-Image + bi-LSTM ResNet +Transformer</td><td>65.43</td><td>56.55</td><td>68.63</td><td>40.06</td><td>57.67</td></tr><tr><td></td><td>66.97</td><td>58.32</td><td>70.10</td><td>43.31</td><td>59.67</td></tr><tr><td>CLIP-Image + Transformer</td><td>70.8</td><td>61.67</td><td>72.43</td><td>47.48</td><td>63.11</td></tr><tr><td>CLIP-It (Gen. Caption)</td><td>74.13</td><td>63.44</td><td>75.86</td><td>50.23</td><td>65.92</td></tr><tr><td>CLIP-It (GT Caption)</td><td>84.98</td><td>71.26</td><td>82.55</td><td>61.46</td><td>75.06</td></tr><tr><td> Unsupervised</td><td>Vid 1</td><td>Vid 2</td><td>Vid 3</td><td>Vid 4</td><td>Avg</td></tr><tr><td>Quasi [46]</td><td>53.06</td><td>53.80</td><td>49.91</td><td>22.31</td><td>44.77</td></tr><tr><td>CLIP-Image + Transformer</td><td>65.44</td><td>57.21</td><td>65.10</td><td>41.63</td><td>57.35</td></tr><tr><td>CLIP-It (Gen. Caption)</td><td>67.02</td><td>59.48</td><td>66.43</td><td>44.19</td><td>59.28</td></tr><tr><td>CLIP-It (GT Caption)</td><td>73.90</td><td>66.83</td><td>75.44</td><td>52.31</td><td>67.12</td></tr></table>",
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+        "text": "Quantitative Results. We compare our method and its ablations to supervised video summarization baselines in Tab.1. We report F1 scores on the TVSum and SumMe datasets for all three data settings. Our full method, CLIP-It (CLIP-Image $+$ Video Caption $^ +$ Transformer), described in Sec. 3, outperforms state-of-the-art by a large margin on all three settings. Particularly, in the Transfer setting we outperform the previous state-of-the-art SumGraph [26] by $5 \\%$ on TVSum and $3 \\%$ on SumMe, indicating that our model is better than the baselines in generalizing to out-of-distribution data. ",
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+        "type": "text",
+        "text": "To prove the effectiveness of each component of our model, we include comparisons to different ablations. In CLIP-Image+Transformer, we ablate the Language-Guided Attention module and directly pass the CLIP-Image features as input to the transformer. As seen, the performance drops by $2 \\%$ indicating that conditioning on CLIP language embeddings leads to better summaries. Substituting the Frame-Scoring Transformer with a Bidirectional LSTM in CLIP-Image+bi-LSTM and CLIP-Image $^ +$ Video Caption+bi-LSTM again results in a performance drop, thus highlighting the need for the Transformer module in our model. For a fair comparison with the baselines, in GoogleNet+Transformer we use the same GoogleNet features provided by Zhang et al. [43] and used by all the other baselines and do not include language features. This method still outperforms Sum",
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+        "img_path": "images/7358c9804a60b78676570b2d26780d98322d1dee29fa09714b106bd270f41b48.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of ground-truth summary to results from CLIP-Image $^ +$ Transformer and the full CLIP-It model (CLIP-Image $^ +$ Video Caption $^ +$ Transformer). The input is a recipe video. Without captions, the model assigns high scores to certain irrelevant frames such as scenes of the woman talking or eating which hurts the precision. With captions, the cross-attention mechanism ensures that frames with important actions and objects are assigned high scores. "
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+        "image_caption": [
+            "Figure 4: Qualitative result comparing the generic summary from CLIP-It with the ground-truth summary. The plots showing predicted and ground-truth frame-level scores are similar, indicating that frames that were given a high score in ground-truth were also assigned high scores by our model. "
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+        "type": "text",
+        "text": "Graph [26]. Replacing the GoogleNet features with ResNet features results in a small performance improvement but CLIP-Image features prove to be most effective. ",
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+        "text": "Tab. 2 compares F1 scores in the unsupervised setting to other unsupervised baselines. CLIP-It (CLIP-Image $+$ Video Caption+Transformer), outperforms the best performing baseline on all settings on both datasets. We again observe large performance improvements in the Transfer setting and notice that overall our unsupervised method performs almost as well as our supervised counterpart. In CLIP-Image+Transformer we ablate the language component which causes a drop in performance, thus proving that language cues are useful in the unsupervised setting as well. Other ablations yield similar results reiterating the need for each component of our method. We also follow Otani et al. [25] and report results on rank based metrics, Kendall’s $\\tau$ [14] and Spearman’s $\\rho$ [49] correlation coefficients in the Supp. ",
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+        "table_caption": [
+            "Table 4: Results on UT Egocentric dataset [16] "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>F-Measure</td><td>Recall</td></tr><tr><td colspan=\"3\">Supervised</td></tr><tr><td>Submod-V+Both et al. [27]</td><td>34.15</td><td>31.59</td></tr><tr><td>CLIP Image + Transformer</td><td>41.58</td><td>39.96</td></tr><tr><td>CLIP-It: CLIP Image + Gen. Video Caption + Transformer</td><td>44.70</td><td>43.28</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption + Transformer</td><td>52.10</td><td>50.76</td></tr><tr><td colspan=\"3\">Unsupervised</td></tr><tr><td>CLIP Image + Transformer</td><td>39.22</td><td>37.46</td></tr><tr><td>CLIP-It: CLIP Image + Gen.Video Caption + Transformer</td><td>42.10</td><td>40.65</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption + Transformer</td><td>49.98</td><td>47.91</td></tr><tr><td colspan=\"3\">Table 5: Results on TV Episodes dataset [16]</td></tr><tr><td>Method</td><td>F-Measure</td><td>Recall</td></tr><tr><td colspan=\"3\">Supervised</td></tr><tr><td>Submod-V+Sem. Rep. et al. [27]</td><td>40.90</td><td>37.02</td></tr><tr><td>CLIP Image+ Transformer</td><td>47.82</td><td>46.02</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption + Transformer</td><td>55.34</td><td>53.90</td></tr><tr><td colspan=\"3\">Unsupervised</td></tr><tr><td>CLIP Image + Transformer</td><td>45.77</td><td>44.01</td></tr><tr><td>CLIP-It: CLIP Image + GT Video Caption+ Transformer</td><td>53.42</td><td>52.50</td></tr></table>",
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+        "type": "text",
+        "text": "Tab. 3 shows F1 scores for the generic setting on the QFVS Dataset [33]. Following [33], we run four rounds of experiments leaving out one video for testing and one for validation, while keeping the remaining two for training. Our method, CLIP-It (Gen. Captions) outperforms both supervised and unsupervised baselines by a large margin on all four videos, and particularly on Video 4, which happens to be the most difficult for all methods. Adding captions helps significantly improve $( 2 \\% )$ the summaries and outperforms the CLIP Image $^ +$ Transformer baseline. To see how well our model would perform if we had perfect captions, we also show results by using the ground-truth captions obtained from VideoSet [40]. Replacing CLIP-Image features with ResNet features causes a drop in performance. Likewise, replacing the Transformer with a bi-LSTM also hurts performance. ",
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+        "type": "text",
+        "text": "We include results of our method (1) without captions (2) using generated captions (3) using the ground truth captions provided by VideoSet [40] for UT Egocentric and TV Episodes datasets in Tables 4 and 5. As ground truth, we obtain 15 summaries for each video using the same greedy n-gram matching and ordered subshot selection procedures as previous work [27]. We follow the same procedure as in prior work [40, 27] for creating and evaluating text summaries from video summaries. Our method outperforms [27] in both the supervised and unsupervised settings on both datasets. ",
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+        "type": "text",
+        "text": "Qualitative Results. We highlight the need to use language, specifically dense video captions, for constructing generic video summaries through a qualitative example in Fig. 3. The input is a video of a woman demonstrating how to make a chicken sandwich. The ground truth summary shows the scores computed by averaging the annotations from all users as in [43] and the keyframes that received high scores. Next we show the result from the baseline CLIP-Image+Transformer, which uses only visual features and no language input. The predicted scores show that the high scoring frames in the ground truth also receive a high score by our baseline, however a lot of irrelevant frames end up receiving a high score too. Thus, when the model finally picks the keyframes it ends up selecting frames where the person is talking or eating the sandwich (shown in red) which do not correspond to the key steps in the video. Adding language guidance via generated captions helps address this problem. The last row shows the captions generated by BMT [11]. The results shown are from our full CLIP-It model. The predicted scores are more similar to ground truth scores and the highest scoring keyframes are the same as the ground truth. ",
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+        "table_caption": [
+            "Table 6: Comparing F1 Scores of different methods on the QFVS dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Vid 1</td><td>Vid 2</td><td>Vid 3</td><td>Vid 4</td><td>Avg</td></tr><tr><td>SeqDPP[51</td><td>36.59</td><td>43.67</td><td>25.26</td><td>18.15</td><td>30.92</td></tr><tr><td>SH-DPP [32]</td><td>35.67</td><td>42.74</td><td>36.51</td><td>18.62</td><td>33.38</td></tr><tr><td>QFVS [33]</td><td>48.68</td><td>41.66</td><td>56.47</td><td>29.96</td><td>44.19</td></tr><tr><td>CLIP-Image + Query + bi-LSTM</td><td>54.47</td><td>48.59</td><td>62.81</td><td>38.64</td><td>51.13</td></tr><tr><td>ResNet + Query + Transformer</td><td>55.19</td><td>51.03</td><td>64.26</td><td>39.47</td><td>52.49</td></tr><tr><td>CLIP-It: CLIP-Image + Query + Transformer</td><td>57.13</td><td>53.60</td><td>66.08</td><td>41.41</td><td>54.55</td></tr></table>",
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+        "image_caption": [
+            "Figure 5: Result of our method on the QFVS dataset. The first row shows some frames from the 4 hour long input video. Given the query \"book and chair\", Summary 1 shows some frames selected by our method. Summary 2 shows frames for the query \"Sun and Tree\". "
+        ],
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+        "type": "text",
+        "text": "Fig. 4 shows compares summaries generated by our method to the ground truth summary on a cooking video from the SumMe dataset. Predicted scores are the frame-wise scores predicted by CLIP-It and GT Scores are the scores computed from user annotations [43]. The top row (True Positives) shows high scoring keyframes which are chosen by both our method and the ground truth, and the green arrows point to the assigned scores. As we see, they are clear, distinct and represent key actions in the recipe. The bottom row (True Negatives) shows frames which are assigned a low score (shown by the red arrows) and are not part of the final summaries (both GT and predicted). E.g., the first frame is irrelevant and corresponds to a segment between key steps, while the second frame has poor lighting and its hard to tell what is being done. ",
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+    {
+        "type": "text",
+        "text": "4.2 Query-Focused Video Summarization ",
+        "text_level": 1,
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+        "text": "In Query-Focused Video Summarization, the summarization process is conditioned on an input user query, thus, multiple summaries can be obtained for the same video using different input queries. ",
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+        "type": "text",
+        "text": "Dataset. We evaluate our method on the QFVS dataset based on the UT Egocentric videos described earlier. The dataset consists of 46 queries for each of the four videos and user-annotated video summaries for each query. ",
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+        "type": "text",
+        "text": "Quantitative Results. In Tab. 6, we compare F1 scores of our method to 3 baselines, SH-DPP [32], Seq-DPP [5], and QFVS [33]. Following [33], we run four rounds of experiments leaving out one video for testing and one for validation, while keeping the remaining two for training. Our full model achieves an avg F1 score of $5 4 . 5 5 \\%$ outperforming the best baseline $( 4 4 . 1 9 \\% )$ by $10 \\%$ . We would like to point out that our method uses more recent image features compared to the baselines. At the same time, when switching from CLIP to ResNet image embedding, we still improve significantly over the baselines. We expect the improvement to also hold with weaker features (e.g., as demonstrated with GoogleNet results in Tab. 1, 2). ",
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+        "type": "text",
+        "text": "Qualitative Results. Fig. 5 shows a result on the QFVS dataset for UT Egocentric videos. The input is an egocentric video shot from a head-mounted camera and spans 4 hours. It consists of multiple events from a person’s day, such as reading books, working on the laptop, walking in the streets, and so on. Summary 1 and 2 show results from our CLIP-It method when the input query is “Book and chair” and “Sun and tree” respectively. The frames shown are frames assigned high-scores by our method. As seen, given the same input video, different queries yield different summaries. ",
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+    {
+        "type": "text",
+        "text": "5 Discussion and Broader Impacts ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We introduced CLIP-It, a unified language-guided framework for generic and query focused video summarization. Video summarization is a relevant problem with many use-cases, and our approach provides greater flexibility to the users, allowing them to guide summarization with open-ended natural language queries. We envision potential positive impact from improved user experience if adopted on video platforms such as YouTube. We rely on an off-the-shelf video captioning model [11] and a large-scale vision-language model (CLIP [29]) which may have encoded some inappropriate biases that could propagate to our model. Our visual inspection of the obtained summarization results did not raise any apparent concerns. However, practitioners who wish to use our approach should be mindful of the sources of bias we have outlined above depending on the specific use case they are addressing. ",
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+        "type": "text",
+        "text": "Acknowledgements. We thank Arun Mallya for very helpful discussions and feedback. We’d also like to thank Huijuan Xu for feedback on the draft. This work was supported in part by DoD including DARPA’s XAI, LwLL, and SemaFor programs, as well as BAIR’s industrial alliance programs. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "type": "text",
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IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2015) 2, 3, 6, 7   \n[37] Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., Anguelov, D., Erhan, D., Vanhoucke, V., Rabinovich, A.: Going deeper with convolutions. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2015) 5   \n[38] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł., Polosukhin, I.: Attention is all you need. Advances in Neural Information Processing Systems (NeurIPS) (2017) 3, 4   \n[39] Wei, H., Ni, B., Yan, Y., Yu, H., Yang, X., Yao, C.: Video summarization via semantic attended networks. The Association for the Advancement of Artificial Intelligence Conference (AAAI) (2018) 3   \n[40] Yeung, S., Fathi, A., Fei-Fei, L.: Videoset: Video summary evaluation through text. arXiv preprint arXiv:1406.5824 (2014) 9   \n[41] Yuan, L., Tay, F.E., Li, P., Zhou, L., Feng, J.: Cycle-sum: Cycle-consistent adversarial lstm networks for unsupervised video summarization. The Association for the Advancement of Artificial Intelligence Conference (AAAI) (2019) 3, 7   \n[42] Zhang, K., Chao, W.L., Sha, F., Grauman, K.: Summary transfer: Examplar-based subset selection for video summarization. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016) 2, 3, 6   \n[43] Zhang, K., Chao, W.L., Sha, F., Grauman, K.: Video summarization with long short-term memory. European Conference on Computer Vision (ECCV) (2016) 2, 3, 4, 6, 7, 9, 10   \n[44] Zhang, K., Grauman, K., Sha, F.: Retrospective encoders for video summarization. European Conference on Computer Vision (ECCV) (2018) 2, 3, 6   \n[45] Zhao, B., Li, X., Lu, X.: Hierarchical recurrent neural network for video summarization. ACM international conference on Multimedia (2017) 3, 5   \n[46] Zhao, B., Xing, E.P.: Quasi real-time summarization for consumer videos. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2014) 7   \n[47] Zhou, K., Qiao, Y., Xiang, T.: Deep reinforcement learning for unsupervised video summarization with diversity-representativeness reward. The Association for the Advancement of Artificial Intelligence Conference (AAAI) (2018) 6   \n[48] Zhou, L., Zhou, Y., Corso, J.J., Socher, R., Xiong, C.: End-to-end dense video captioning with masked transformer. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2018) 3   \n[49] Zwillinger, D., Kokoska, S.: Crc standard probability and statistics tables and formulae. CRC Press (1999) 8 ",
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parse/train/784_F-WCW46/784_F-WCW46.md

@@ -0,0 +1,201 @@
+# RETHINKING SAMPLING IN 3D POINT CLOUD GENERATIVE ADVERSARIAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this paper, we examine the long-neglected yet important effects of point sampling patterns in point cloud GANs. Through extensive experiments, we show that sampling-insensitive discriminators (e.g. PointNet-Max) produce shape point clouds with point clustering artifacts while sampling-oversensitive discriminators (e.g. PointNet++, DGCNN, PointConv, KPConv) fail to guide valid shape generation. We propose the concept of sampling spectrum to depict the different sampling sensitivities of discriminators. We further study how different evaluation metrics weigh the sampling pattern against the geometry and propose several perceptual metrics forming a sampling spectrum of metrics. Guided by the proposed sampling spectrum, we discover a middle-point sampling-aware baseline discriminator, PointNet-Mix, which improves all existing point cloud generators by a large margin on sampling-related metrics. We point out that, given that recent research has been focused on the generator design, the discriminator design needs more attention. Our work provides both suggestions and tools for building future discriminators. We will release the code to facilitate future research.
+
+# 1 INTRODUCTION
+
+Point cloud, as the most common form of 3D sensor data, has been widely used in a variety of 3D vision applications due to its compact yet expressive nature and its amenability to geometric manipulations. It is natural to consider how to generate point cloud through deep learning approaches, which has been a popular research topic recently. The previous research efforts in the community have been mainly devoted to conditional generation of point clouds with 3D supervision. The condition could either be images (Fan et al., 2017; Groueix et al., 2018; Park et al., 2019) or partial point clouds (Li et al., 2019; Yang et al., 2018).
+
+![](images/c2faa45a97279dcce0b62c1f7a939ffd1317175ba945bb4c981516043c089985.jpg)  
+Figure 1: We visualize the behavior of different discriminators when judging four different chair point clouds. (a) and (b) are generated results, (c) and (d) are point clouds sampled using FPS and uniform sampling from a real chair surface. When training the discriminators on data like (d), different discriminators make distinct decisions on the point cloud realness forming a sampling spectrum ranging from sampling-insensitive, sampling-aware to sampling-oversensitive. We advocate samplingaware discriminators in the middle of the spectrum, which provide good guidance for fixing geometric flaws (a) and big sample artifacts (b) and tolerate subtle sampling differences between (c) and (d).
+
+Generating 3D point clouds with GANs in an unsupervised manner is an important but less explored problem. 3D point cloud GAN learns to transform a random latent code into a 3D surface point cloud by playing an adversarial game. Its development is still in an early stage compared with 2D image GANs. While existing works such as (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019) have developed a variety of generators, they all use PointNet (Qi et al., 2017a) with max pooling (PointNet-Max) as their discriminator. PointNet, which is essentially a pointwise MLP followed by a global pooling operation, is too limited in capturing shape details for a successful GAN. However, advanced networks, e.g. PointNet++(Qi et al., 2017b), DGCNN(Wang et al., 2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019), which leverage relative positions between points and hierarchical feature extraction, may not help. From our empirical study, we find they all fail to be a functioning discriminator. Understanding their failure mechanism and improving discriminator design are hence important and urgent.
+
+To design a better discriminator, we first need to answer the following question: what should the discriminator examine for improving the generation quality? Or, even more fundamentally, what does it mean by the quality of generated point clouds?
+
+Since a shape point cloud are the points sampled from an object surface, its quality should be evaluated from two perspectives: the depicted surface geometry and the point sampling. Arguably, geometry plays a decisive role and should be the main focus of a discriminator. However, when the generated point clouds have a good shape, there is still a full spectrum on how much a discriminator cares about the sampling patterns. We introduce the concept of Sampling Spectrum to depict the sampling sensitivity of discriminators, as illustrated in Figure 1. A sampling-insensitive discriminator (e.g. PointNet-Max) may ignore the point density variations as long as it perceives a good overall shape. Such a discriminator could identify big geometric flaws as shown in Figure 1 (a), but turns a blind eye to highly non-uniform density distribution, e.g. point clusters in Figure 1 (b). On the other extreme, a sampling-oversensitive discriminator (e.g. PointNet $^ { + + }$ , DGCNN, KPConv) can even tell the subtle difference in sampling patterns, e.g. between furthest point sampling (FPS) in Figure 1 (c) and uniform sampling in Figure 1 (d), and hence can be very narrow-minded about what a real point cloud should look like. A sampling-aware discriminator, e.g. PointNet-Mix/Attention, which lies in the middle of the spectrum, is able to identify density-related artifacts such as point clusters in Figure 1 (b), while not being too sensitive to different sampling patterns, such as Figure 1 (c) and (d).
+
+Resembling the sampling spectrum of discriminators, we also examine the existing point-cloud GAN evaluation metrics from the perspective of sampling sensitivity and propose several perceptual metrics forming a Sampling Spectrum of evaluation metrics. Understanding how the metrics weigh between sampling and geometry is a prerequisite for evaluating point cloud GANs. Many of the existing sampling-insensitive metrics only evaluate the geometry factor of the generated point clouds shapes, which are blind to the obvious point clustering artifacts and uneven point density. We propose novel sampling-sensitive metrics to further complete the spectrum of point-cloud GAN evaluation metrics.
+
+Guided by the proposed sampling spectrum of discriminators and evaluation metrics, experiments show that different discriminators in the spectrum could provide very different suggestions to improve a generator according to its sampling sensitivity. Sampling-insensitive discriminators, e.g. PointNetMax, is unaware of point density variations and hence its generated point cloud inevitably suffer from clustering artifacts, while sampling-oversensitive discriminators, e.g. PointNe $^ { + + }$ , DGCNN, KPConv, PointConv, simply fail to function as the discriminators and can generate much degraded point cloud shapes. We design a diagnostic “no generator” experiment to factor out the impact from generators and reveal that the gradients of sampling-oversensitive discriminators prioritize adjusting sampling patterns over producing better shape geometry. Picking a middle-point on the sampling spectrum, we discover a simple yet effective sampling-aware discriminator, PointNet-Mix, and find that it can supervise both shape generation and point density uniformity. It improves all existing generators by a large margin on sampling-related metrics. Surprisingly, we find that even the most naive fully-connected generator, coupled with PointNet-Mix, simply beats all the start-of-the-art point cloud GANs. This discovery conveys an important message to the community: instead of focusing on the generator design, people should invest more time into discriminator and seek for more powerful sampling-aware discriminators.
+
+# 2 POINT CLOUD GAN LANDSCAPE
+
+In this section, we review the current state of point cloud GAN covering the generators, the discriminators, and the evaluation metrics we are examining in this work.
+
+# 2.1 POINT CLOUD GAN GENERATORS
+
+Recent point cloud GAN works primarily focus on the generator design. The generator takes input a random noise $\cdot$ and outputs a point cloud $\boldsymbol { p } \in \mathbb { R } ^ { N \times 3 }$ . The existing generators can be categorized into two classes: fully-connected (FC) generators and graph convolutional generators. The first point cloud GAN, r-GAN (Achlioptas et al., 2018), simply uses an FC network as its generator. GraphCNN-GAN (Valsesia et al., 2018) and TreeGAN (Shu et al., 2019) are the rest two published works in this field that use graph convolutional generators. The two methods are very similar in principle. The main difference lies in how they build graphs. GraphCNN-GAN builds a dynamic $k$ -nn graph based upon feature space distance while TreeGAN enforces a tree structure throughout its sequential graph expansion and the messages can only be passed from ancestors vertices to descendants vertices.
+
+Deformation-based decoders (Groueix et al., 2018; Yang et al., 2018) are widely used in the point cloud auto-encoder networks for 3D shape reconstruction. The decoders leverage Multiple-layer Perceptrons (MLP) to deform template surfaces into shape surfaces taking as inputs the concatenation of template point coordinates and the latent shape feature vectors. Though the decoders can truly act as generators for point cloud GANs, they have not yet been used in unconditioned point cloud GAN literature. Recently, Mo et al. (Mo et al., 2020) use deformation-based decoder as part generators for structure-conditioned point cloud GAN.
+
+# 2.2 POINT CLOUD GAN DISCRIMINATORS
+
+All existing works on unconditioned point cloud GANs use PointNet with max-pooling (PointNetMax) as their discriminators. The discriminator takes input a point cloud $p \in \mathbb { R } ^ { N \times 3 }$ and outputs a score from 0 to 1. PointNet learns a function $h$ that maps each point $p _ { i }$ in the point cloud to a per-point feature $h ( p _ { i } ) \in \mathbb { R } ^ { d }$ and then extracts a permutation-invariant global feature $F \in \mathbb { R } ^ { d }$ by pooling the per-point features across all points using a symmetric function $g$ , which can be max pooling, average pooling, etc. Namely, we have $F = \overline { { g } } ( \{ \dot { h ( p _ { 1 } ) } , h ( p _ { 2 } ) , \cdots , h ( \overline { { p _ { N } } } ) \} )$ .
+
+PointNet-Max/Avg. Though $g$ can be any symmetric function, most existing works use maxpooling. In PointNet, the authors show that max-pooling outperforms average-pooling on 3D shape classification tasks. They further show that the global feature $F$ obtained from max-pooling is determined by only a sparse subset of the points, namely critical points $\mathcal { C } _ { S }$ , bringing PointNet-Max with robustness against small data perturbation and corruption. However, this property may limit its discriminative power on telling the density variations and classifying different sampling patterns. To investigate how different aggregation operations affect the sampling sensitivity and generation quality, we study the two common choices of the symmetric function $g$ , max-pooling, and average-pooling, in this paper. We will show in Sec. 3 that using different aggregation operations makes huge differences when adapting PointNet-based networks as point cloud GAN discriminators.
+
+PointNet-Mix. By simply concatenating the max-pooling feature and the average pool feature, we obtain another permutation-invariant feature. We name this PointNet-Mix. Formally, $F _ { \mathrm { m i x } } =$ $[ \operatorname* { m a x } \{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \}$ ; $\arg \{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \} ] \in \mathbb { R } ^ { 2 d }$ . The mix-pooling operation is a special choice of the symmetric function $g$ . We will show in our experiments that PointNet-Mix, though simple, surprisingly improves the performance for most point cloud GANs by a large margin on sampling-aware metrics.
+
+PointNet-Attention. A recent point cloud upsampling work (Li et al., 2019) incorporates a selfattention module into PointNet for its discriminator and shows improved point density in the upsampled point cloud. We denote this discriminator as PointNet-Attention. The self-attention module learns three separate MLPs to transform each $h ( p _ { i } )$ into $f ( p _ { i } ) , l ( p _ { i } ) , k ( p _ { i } ) \in \mathbb { R } ^ { d }$ correspondingly. Then an attention weight matrix is formed by $W =$ SoftMax $\left( [ f ( p _ { 1 } ) , . . . , f ( p _ { N } ) ] ^ { T } [ ( l ( p _ { 1 } ) , . . . , l ( p _ { N } ) ) ] \right) \ \in$ $\mathbb { R } ^ { N \times N }$ . A weighted features is then obtained through $w ( p _ { i } ) \ = \ h ( p _ { i } ) \ + \ W ^ { T } k ( p _ { i } )$ . The final aggregated features of PointNet-Attention is a max-pooling of the weighted feature, namely $F _ { \mathrm { a t t e n t i o n } } = \operatorname* { m a x } \{ w ( p _ { 1 } ) , . . . , w ( p _ { N } ) \}$ . Note that PointNet-Attention allows the per-point features to communicate with each other according to their similarity, which is more sensitive to sampling patterns than PointNet-Max. We investigate this strategy in our paper as well.
+
+Discriminators beyond PointNet. Recently, there have been many works (Qi et al., 2017a; Li et al., 2018; Wang et al., 2019; Hermosilla et al., 2018; Thomas et al., 2019) extending PointNet to more advanced 3D deep learning architectures on point clouds. They improve PointNet by extracting more local or hierarchical geometric features via point cloud convolutions or point cloud graph learning. Though proven to be effective on shape classification and segmentation tasks, no published work examines adapting them as point cloud GAN discriminators. In this paper, we investigate four exemplar beyond-PointNet discriminators: PointNet $^ { + + }$ (Qi et al., 2017b), DGCNN (Wang et al., 2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019).
+
+# 2.3 POINT CLOUD GAN EVALUATION METRICS
+
+Achlioptas et al. (Achlioptas et al., 2018) introduces two distance metrics in the Euclidean space for evaluating point cloud GAN: the coverage scores (COV) computing the fraction of the point clouds in $B$ that are closed to the point clouds in $A$ using either Earth Mover’s distance (EMD) or Chamfer distance (CD), and the Minimum matching distance (MMD) scores measuring the fidelity of $A$ with respect to $B$ using either EMD or CD. In the field of 2D image GAN, it is common to use perceptual metrics, such as the Frechét distance (Heusel et al., 2017) between real and fake Gaussian measures in the feature spaces, for evaluating the generated image results. Formally, Frechet Distance $= | | \mu _ { r } - \mu _ { g } | | ^ { 2 } + \operatorname { T r } ( \Sigma _ { r } + \Sigma _ { g } - 2 \left( \Sigma _ { r } \Sigma _ { g } \right) ^ { 1 / 2 } )$ , where $\mu$ and $\Sigma$ are the mean vector and the covariance matrix of the features calculated from either real or fake data distribution, and $\mathrm { T r }$ is the matrix trace. For point clouds, Shu et al. (Shu et al., 2019) proposes Frechét Point Cloud Distance (FPD), which uses the features extracted from a pre-trained PointNet-Max model. In this paper, we position the existing metrics on a sampling spectrum according to their sampling sensitivity and propose novel sampling-aware metrics to augment the spectrum.
+
+# 3 SAMPLING SPECTRUM OF DISCRIMINATORS
+
+While recent works propose many advanced generator improvements for point cloud GANs, we find that designing a good discriminator is of equal importance, if not more. In this section, we introduce the sampling spectrum of discriminators, on which we thoroughly examine the positions of different discriminators that explain their behaviors when training point cloud GANs.
+
+# 3.1 SAMPLING SENSITIVITY OF DISCRIMINATORS
+
+The sampling sensitivity of a discriminator depicts how much it responds to a change in the point density or sampling pattern of an input point cloud. We find it extremely hard to quantitatively measure this sensitivity given the difficulty of measuring changes in the sampling patterns. Naively using Euclidean metrics (e.g. CD or EMD) to measure the distance between two sampled point sets is not a solution, since given the same distance budget, the discriminator’s responses can be dramatically different depending on how the sampled points move. Instead, we can set landmarks in the continuous spectrum of sampling sensitivity by examining the discriminative power of the discriminators against different sampling patterns under a series of experiments from easy to hard. Specifically, we design two experiments to test whether a discriminator could tell clustering artifacts in point clouds and whether it could distinguish between FPS and uniform sampling patterns. Accordingly, we divide the spectrum into three regimes: sampling-insensitive/-aware/-oversensitive, as shown in Fig.1.
+
+A sampling-insensitive discriminator does not respond to local point density changes if the overall shape remains roughly the same. This kind of discriminators can’t tell clustering artifacts, i.e. Fig.1 (b), and thus may cause the non-uniform density in the generated point clouds.
+
+A sampling-aware discriminator can notice the significant point non-uniformity and is hence capable of supervising the generator to enforce a similar sampling distribution to the training data, while being ignorant to subtle changes when the sampling is already uniform in an intermediate scale. Such discriminator will judge Fig.1 (b) as fake but can’t tell the difference between Fig.1 (c) and (d).
+
+A sampling-oversensitive discriminator can tell very subtle difference in the sampling patterns, e.g.   
+between FPS and uniform sampling, even if the underlying shapes are the same (Fig.1 (c) and (d)).
+
+# 3.2 SAMPLING SENSITIVITY EXAMINATION RESULTS
+
+We design diagnostic experiments to categorize point cloud GAN discriminators on the sampling spectrum: sampling-insensitive (PointNet-Max), sampling-aware (PointNet-Avg, PointNet-Mix, PointNet-Attention (Li et al., 2019)), and sampling-oversensitive (PointNet+ $^ { - + }$ , DGCNN, KPConv, PointConv).
+
+The Discriminating Power against Clustering Artifacts. To examine whether the discriminators can tell clustering artifacts or not, we create a diagnostic classification dataset using the models from ShapeNet (Chang et al., 2015). Taking 100 chairs, we uniformly sample 2048 points from each shape, forming a set of real point clouds. To form a fake set of point clouds, for each chair, we first uniformly sample 1024 points, and then densely sample another 1024 points around a random position on the chair within a 0.1 radius. The real/fake point clouds are used as the labeled training dataset. We repeat the same process to generate a test dataset using a different set of 100 chairs.
+
+<table><tr><td>Experiment</td><td>PN-Max</td><td>PN-Avg</td><td>PN-Mix</td><td>PN-Att</td><td>PN++</td><td>DGCNN</td><td>PointConv</td><td>KPConv</td></tr><tr><td>Clustering art.</td><td>56%/53%</td><td>98%/99%</td><td>97%/96%</td><td>94%/93%</td><td>100%/98%</td><td>100%/99%</td><td>100%/99%</td><td>100%/99%</td></tr><tr><td>FPS vs.uniform</td><td>50%/50%</td><td>50.3%/50%</td><td>51.5%/50%</td><td>50.1%/50%</td><td>100%/97%</td><td>100%/96%</td><td>100%/98%</td><td>100%/99%</td></tr><tr><td>ESS</td><td>0.03</td><td>0.49</td><td>0.46</td><td>0.43</td><td>0.95</td><td>0.95</td><td>0.97</td><td>0.99</td></tr></table>
+
+Table 1: Evaluating the discriminating power of the discriminators against clustering artifacts and sutble change in sampling patterns along with their empirical sampling sensitivity (ESS). In each cell of the top two rows, the left shows the training accuracy while the right shows the test accuracy.
+
+We supervisely train each discriminator to classify the real/fake point clouds. We train them 200 epochs until convergence. The training and test accuracies are shown in the first row of Table 1. Despite the huge density variation and the remarkable clustering artifacts, PointNet-Max is just barely better than a random guess while the rest of the discriminators are very successful in telling the fake from the real. This indicates that PointNet-Max is sampling-insensitive. Note that the discriminating power of a network towards certain artifacts is maximized under such supervised training scheme. A network will fail to identify such artifacts in an adversarial training scheme if it fails in a supervised training scheme. We will see in Sec. 5.2, point clouds generated by GAN using PointNet-Max as the discriminator indeed suffer from non-uniform density artifacts.
+
+Our insight why PointNet-Avg/Mix can tell the artifacts but PointNet-Max fails is that the average pooling feature computes the center of the mass of the points in feature space and is hence aware to certain global non-uniform density distributions. PointNet-Attention leverages a learnable weighted averaging and is hence capable to identify the difference.
+
+Distinguishing between FPS and Uniform Sampling. We construct another diagnostic dataset with real and fake data which are the same in their shapes but only differ in their sampling patterns. Specifically, we perform uniform sampling to generate real data and use FPS to generate fake data from 100 chairs. Figure 1 (c) and (d) illustrate the different sampling pattern outcomes.
+
+We present the training and test accuracies in the second row of Table 1 and show that PointNet $^ { + + }$ , DGCNN, KPConv, and PointConv can perfectly distinguish the subtle difference in sampling patterns while the rest discriminators make no progress even on the training set. The experiment indicates that PointNet-Avg/Mix/Attention are sampling-aware while PointNet++, DGCNN, KPConv, and PointConv are sampling-oversensitive.
+
+We believe, for PointNet+ $^ +$ , DGCNN, KPConv, and PointConv, their capability of distinguishing FPS from uniform sampling owes to their usage of relative point positions or edge information, which are highly sensitive to any change in sampling. We will show in Sec. 5.3 that their remarkable discriminating power on sampling patterns actually leads to the failures as functioning discriminators to train point cloud GANs.
+
+Empirical Definition of Sampling Sensitivity Given the difficulty of measuring the change in sampling pattern, we would like to provide an empirical definition of sampling sensitivity of one discriminator according to its performance on the two experiments above: when the discriminator fails in telling clustering artifacts and telling the difference between FPS and uniform sampling pattern, its sampling sensitivity should be closed to 0; when the discriminator can tell clustering artifacts but can’t distinguish between FPS and uniform sampling, we want to assign it a sampling sensitivity closed to 0.5 ; when the discriminator can do both, we want to assign a sampling sensitivity closed to 1 to it. Based on this, we define empirical sampling sensitivity $\cdot$ as:
+
+ESS = Acc(Tell clustering art.) $^ +$ Acc(Tell FPS from uniform sampling) $^ { - 1 }$ , where Acc denotes the test accuracy of an experiment. See Table 1 for results. Using this metric, we would form a continuous sampling sensitivity spectrum.
+
+# 4 SAMPLING SPECTRUM OF EVALUATION METRICS
+
+Similar to the discriminator design, it is very important to understand how different evaluation metrics weigh the differences in sampling patterns against geometry quality. Thus, we introduce the sampling spectrum of evaluation metrics, which exactly resembles the sampling spectrum of discriminators introduced in Sec. 3. On the spectrum, we have sampling-insensitive metrics that measure only the shape geometry and are ignorant of the sampling patterns, and sampling-sensitive metrics that measure both at the same time.
+
+For perceptual metrics, i.e. Frechét distances in feature spaces, the sampling sensitivity of the metric purely depends on the sampling sensitivity of its feature extractor. Frechét distance measured in different feature spaces may respond very differently to changes in point density and sampling patterns. In this work, we examine three Frechét distance metrics that extract features using PointNetMax, PointNet-Mix, and DGCNN, respectively. We denote them as Frechét PointNet-Max Distance (FPD-Max), Frechét PointNet-Mix Distance (FPD-Mix), and Frechét DGCNN Distance (FGD). We pretrain all the three feature extraction networks on ModelNet40 (Wu et al., 2015) shape classification.
+
+Table 2: Examining sampling sensitivity of evaluation metrics.   
+
+<table><tr><td>Data</td><td>FPD-Mix↓</td><td>FPD-Max↓</td><td>FGD↓</td><td>MMD-E↓</td><td>MMD-C↓</td><td>COV-E↑</td><td>COV-C↑</td></tr><tr><td>Uniformre-sampling</td><td>0.1153</td><td>0.0926</td><td>0.8141</td><td>0.1104</td><td>0.00145</td><td>70.69</td><td>72.16</td></tr><tr><td>Farthest point sampling</td><td>0.1700</td><td>0.1558</td><td>1.8833</td><td>0.1064</td><td>0.00137</td><td>67.74</td><td>69.36</td></tr><tr><td>Biased sampling</td><td>2.8631</td><td>0.3524</td><td>9.6719</td><td>0.2469</td><td>0.00145</td><td>23.12</td><td>71.28</td></tr></table>
+
+To examine the sampling sensitivity of FPD-Mix/Max and FGD, we create several copies of the training split of our ShapeNet chair dataset (see Sec.5.1), each of which uses a different sampling strategy to obtain the shape point clouds. The reference one used as the ground truth is using uniform sampling. Then we consider 1) uniform sampling with a different random seed; 2) FPS; and 3) biased sampling with clustered artifacts (as described in Sec. 3.2). We use all the available metrics to evaluate their distances to the ground truth data. The results are shown in Table 2.
+
+We observe that the Frechét distance metrics share the same sampling sensitivity of their corresponding discriminators. For example, since PointNet-Max is sampling-insensitive, FPD-Max remains very low even on biased sampling data, hence FPD-Max serves as a perceptual geometry metric, which is ignorant to sampling patterns. Similarly, we find that FPD-Mix is sampling-aware since it clearly detects the biased sampling patterns while not being able to distinguish the uniform sampling and FPS, while FGD is sampling-oversensitive in that it can tell apart FPS and uniform sampling.
+
+For Euclidean distance metrics, results in Table 2 show that COV-EMD and MMD-EMD are samplingaware, which is intuitively reasonable since EMD enforces a one-to-one matching and is aware of the point density, while COV-CD and MMD-CD are sampling-insensitive.
+
+# 5 EXPERIMENTS
+
+Aware of both the sampling spectrums, we conduct experiments to further evaluate the performance of point cloud GANs under various evaluation metrics. We show that the point cloud GANs using sampling-insensitive discriminators may produce point clustering artifacts, while samplingoversensitive discriminators fail to supervise point cloud GAN training at all. We further devise a diagnostic "no-generator" experiment that factors out the generators to better illustrate our discoveries on discriminators. More interesting, we find that the simple PointNet-Mix paired of any generator, even with the most naive fully-connected one, achieves the state-of-the-art performance.
+
+# 5.1 SETTING AND DATASETS
+
+We provide a thorough comparison of all the discriminators investigated in Sec.3 combining with all the available generators in the published literature, including the FC generator proposed in rGAN (Achlioptas et al., 2018), and graph convolutional generators used in TreeGAN (Shu et al., 2019). We also add a deformation-based generator into the comparison given its popularity for supervised point cloud reconstruction (Groueix et al., 2018; Yang et al., 2018).
+
+We use two datasets to evaluate the GANs. One is a single-category dataset containing point clouds sampled from all 6,778 chair meshes in ShapeNet (Chang et al., 2015). The other is a multi-category dataset combining shapes from airplane, car, chair, rifle, sofa, table, vessel categories in ShapeNet. The multi-category dataset contains 34,313 shapes in total. We uniformly sample 2048 points from each shape to form the two datasets. We follow the $8 5 \% / 5 \% / 1 0 \%$ train/validation/test split in (Achlioptas et al., 2018) and use WGAN-gp (Gulrajani et al., 2017) for the GAN training, similar to previous works (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019).
+
+# 5.2 EVALUATING POINTNET-BASED DISCRIMINATORS WITH VARIOUS GENERATORS
+
+We report the performance for point cloud GANs that combine PointNet-Max/Min/Attention discriminators and FC/Deform/TreeGAN/Graph-CNN generators in Table 3. We observe that GANs using PointNet-Mix as the discriminator outperform the ones using PointNet-Max/Attention across all different generators. On sampling-aware/sensitive metrics (i.e. FPD-Mix, FGD, MMD-EMD, COV-EMD), PointNet-Mix is always significantly better than PointNet-Max and is better than PointNet-Attention mainly on FPD-Mix and FGD. Regarding geometry quality evaluated using sampling-insensitive metrics (i.e. FPD-Max, MMD-CD, COV-CD), PointNet-Mix is always significantly better than PointNet-Attention on FPD-Max and COV-CD while remaining on a par with PointNet-Max.
+
+Table 3: Evaluating PointNet-based discriminators with different generators. We observe that PointNet-Mix can significantly improve all sampling-aware/sensitive metric, including FPD-Mix, FGD, MMD-EMD, COV-EMD for all the generators. When paired with FC generator, PointNet-Mix achieves the best performance. We include PointFlow(Yang et al., 2019) and Oracle for a comparison.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Generator</td><td rowspan=1 colspan=1>Pooling</td><td rowspan=1 colspan=1>FPD-Mix↓</td><td rowspan=1 colspan=1>FPD-Max↓</td><td rowspan=1 colspan=1>FGD↓</td><td rowspan=1 colspan=1>MMD-E↓</td><td rowspan=1 colspan=1>MMD-C↓</td><td rowspan=1 colspan=1>COV-E↑</td><td rowspan=1 colspan=1>COV-C↑</td></tr><tr><td rowspan=10 colspan=1>Chair</td><td rowspan=2 colspan=1>FCFC</td><td rowspan=2 colspan=1>MaxMix</td><td rowspan=2 colspan=1>1.5710.184</td><td rowspan=2 colspan=1>0.2110.209</td><td rowspan=1 colspan=1>7.030</td><td rowspan=1 colspan=1>0.1017</td><td rowspan=1 colspan=1>0.00164</td><td rowspan=1 colspan=1>23.56</td><td rowspan=1 colspan=1>72.75</td></tr><tr><td rowspan=1 colspan=1>2.124</td><td rowspan=1 colspan=1>0.0674</td><td rowspan=1 colspan=1>0.00196</td><td rowspan=1 colspan=1>73.64</td><td rowspan=1 colspan=1>74.96</td></tr><tr><td rowspan=7 colspan=1>FCDeformDeformDeformTreeGANTreeGANGraph-CNN</td><td rowspan=1 colspan=1>Attention</td><td rowspan=1 colspan=1>0.635</td><td rowspan=1 colspan=1>0.672</td><td rowspan=1 colspan=1>4.971</td><td rowspan=1 colspan=1>0.1156</td><td rowspan=1 colspan=1>0.00160</td><td rowspan=1 colspan=1>68.92</td><td rowspan=1 colspan=1>70.54</td></tr><tr><td rowspan=2 colspan=1>MaxMix</td><td rowspan=2 colspan=1>0.9130.534</td><td rowspan=2 colspan=1>0.2680.373</td><td rowspan=1 colspan=1>5.602</td><td rowspan=2 colspan=1>0.09080.0695</td><td rowspan=2 colspan=1>0.002010.00200</td><td rowspan=2 colspan=1>68.576.29</td><td rowspan=2 colspan=1>72.6175.11</td></tr><tr><td rowspan=1 colspan=1>2.836</td></tr><tr><td rowspan=4 colspan=1>AttentionMaxMixMax</td><td rowspan=1 colspan=1>0.696</td><td rowspan=1 colspan=1>0.755</td><td rowspan=1 colspan=1>2.987</td><td rowspan=1 colspan=1>0.1141</td><td rowspan=1 colspan=1>0.00160</td><td rowspan=1 colspan=1>68.77</td><td rowspan=1 colspan=1>69.36</td></tr><tr><td rowspan=3 colspan=1>1.4420.2931.034</td><td rowspan=1 colspan=1>0.654</td><td rowspan=2 colspan=1>7.8084.032</td><td rowspan=2 colspan=1>0.09620.0704</td><td rowspan=2 colspan=1>0.001910.00211</td><td rowspan=3 colspan=1>24,7474.8248.90</td><td rowspan=3 colspan=1>73.4978.7963.18</td></tr><tr><td rowspan=1 colspan=1>0.334</td></tr><tr><td rowspan=1 colspan=1>0.981</td><td rowspan=1 colspan=1>7.494</td><td rowspan=1 colspan=1>0.0812</td><td rowspan=1 colspan=1>0.00191</td></tr><tr><td rowspan=1 colspan=2>PointFlowOracle</td><td rowspan=1 colspan=1>1.7820.088</td><td rowspan=1 colspan=1>1.4050.093</td><td rowspan=1 colspan=1>3.2560.814</td><td rowspan=1 colspan=1>0.13350.0594</td><td rowspan=1 colspan=1>0.001900.00165</td><td rowspan=1 colspan=1>71.7679.56</td><td rowspan=1 colspan=1>71.7379.69</td></tr><tr><td rowspan=9 colspan=1>Multi-Cat</td><td rowspan=3 colspan=1>FCFCFC</td><td rowspan=2 colspan=1>MaxMix</td><td rowspan=1 colspan=1>1.553</td><td rowspan=1 colspan=1>0.354</td><td rowspan=1 colspan=1>6.981</td><td rowspan=1 colspan=1>0.0842</td><td rowspan=1 colspan=1>0.00153</td><td rowspan=1 colspan=1>35.16</td><td rowspan=1 colspan=1>64.16</td></tr><tr><td rowspan=1 colspan=1>0.255</td><td rowspan=1 colspan=1>0.285</td><td rowspan=1 colspan=1>2.550</td><td rowspan=1 colspan=1>0.0656</td><td rowspan=2 colspan=1>0.001840.00134</td><td rowspan=2 colspan=1>73.572.19</td><td rowspan=2 colspan=1>72.1669.63</td></tr><tr><td rowspan=1 colspan=1>Attention</td><td rowspan=1 colspan=1>0.414</td><td rowspan=1 colspan=1>0.453</td><td rowspan=1 colspan=1>4.234</td><td rowspan=1 colspan=1>0.1188</td></tr><tr><td rowspan=5 colspan=1>DeformDeformDeformTreeGANTreeGAN</td><td rowspan=2 colspan=1>MaxMix</td><td rowspan=1 colspan=1>1.072</td><td rowspan=1 colspan=1>0.633</td><td rowspan=1 colspan=1>3.845</td><td rowspan=1 colspan=1>0.0799</td><td rowspan=1 colspan=1>0.00179</td><td rowspan=1 colspan=1>62.5</td><td rowspan=1 colspan=1>64.5</td></tr><tr><td rowspan=1 colspan=1>0.614</td><td rowspan=1 colspan=1>0.349</td><td rowspan=1 colspan=1>2.451</td><td rowspan=1 colspan=1>0.0670</td><td rowspan=1 colspan=1>0.00191</td><td rowspan=1 colspan=1>70.83</td><td rowspan=2 colspan=1>68.8369.60</td></tr><tr><td rowspan=1 colspan=1>Attention</td><td rowspan=1 colspan=1>0.616</td><td rowspan=1 colspan=1>0.720</td><td rowspan=1 colspan=1>2.531</td><td rowspan=1 colspan=1>0.1113</td><td rowspan=1 colspan=1>0.00141</td><td rowspan=1 colspan=1>72.04</td></tr><tr><td rowspan=2 colspan=1>MaxMix</td><td rowspan=2 colspan=1>1.7140.388</td><td rowspan=1 colspan=1>0.437</td><td rowspan=1 colspan=1>6.342</td><td rowspan=1 colspan=1>0.1093</td><td rowspan=1 colspan=1>0.00158</td><td rowspan=2 colspan=1>25.3372.66</td><td rowspan=2 colspan=1>67.071.0</td></tr><tr><td rowspan=1 colspan=1>0.420</td><td rowspan=1 colspan=1>4.300</td><td rowspan=1 colspan=1>0.0699</td><td rowspan=1 colspan=1>0.00184</td></tr><tr><td rowspan=1 colspan=2>Oracle</td><td rowspan=1 colspan=1>0.131</td><td rowspan=1 colspan=1>0.177</td><td rowspan=1 colspan=1>0.673</td><td rowspan=1 colspan=1>0.06012</td><td rowspan=1 colspan=1>0.00124</td><td rowspan=1 colspan=1>77.14</td><td rowspan=1 colspan=1>77.57</td></tr></table>
+
+In Figure 2, we present the generated point clouds for all the experiments with color-coding for the local point density. We see that the generators trained using PointNet-Max usually suffer from non-uniform density, except for the deformation generator. On the chair class, points are usually clustering around the seat area while being sparse at the back. On the contrary, PointNet-Mix enforces a globally uniform point density and hence greatly improves the visual quality of the generated point clouds. PointNet-Attention is in the between.
+
+With PointNet-Mix, we observe that the most naive FC generator works the best outperforming the previously state-of-the-state generators on almost of metrics. This showcases the importance of being sampling-aware but not sampling-insensitive/oversensitive as a discriminator. It also suggests that future works may focus on designing more powerful sampling-aware discriminators.
+
+# 5.3 DIAGNOSING FAILURES FOR VARIOUS DISCRIMINATORS
+
+Observing that the discriminator forms the bottleneck of the point cloud GAN, it is important to examine whether advanced networks, e.g. PointNet+ $^ +$ (Qi et al., 2017b), DGCNN (Wang et al., 2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019) can outperform PointNet-based discriminator.
+
+Surprisingly, we fail to observe any of them ever to be successful to train a GAN in our extensive experiments, in which we perform a systematic search, including varying generator architecture, changing hyperparameters (e.g. the number of layers and parameters in discriminator, learning rate), whether use batch norm, etc. During GAN training, we observe the four advanced discriminator behave very similarly: they are very discriminative in the sense that the gaps between the real and fake scores remain significantly large, but the quality of the generated point clouds stays very bad. The contradictory behaviors seem unreasonable at the first glance: how can a discriminator be very discriminative but teach nothing to the generator? Thus, we design a diagnostic "no generator" experiment to examine the underlying reasons.
+
+No Generator Experiments. During GAN training, the gradients from discriminator output scores back-propagate to the generated point clouds and then further back-propagate to the weights of the generator to update the generator. We design a diagnostic experiment that removes the generator and examines whether the discriminator gradients are informative enough for supervising point cloud GANs.
+
+Concretely, we take $\cdot$ real chairs as real data and we randomly initialize $M$ learnable point clouds $p \in \mathbb { R } ^ { N \times 3 }$ with i.i.d Gaussian noises $\cdot$ . For each training iteration, we sample $\cdot$ point clouds from the real dataset as the real data and sample $\cdot$ point clouds from the learnable point
+
+![](images/9ab46f0bd251aa0ef4577907186cc5ad8e8d73246239da39a1cce07f60003cd8.jpg)
+
+![](images/e359eb374e57942b0203cde57d30eb2a3b5555e59ff8408d26bfd94052794c2d.jpg)  
+Figure 2: Visualization of point clouds generated by different methods. We show generated point clouds in few exemplar shapes for a fair and easier comparison regarding their sampling quality and geometry quality. We color-code the local point density that ranges from sparse (dark blue) to dense (light yellow).   
+Figure 3: The diagnostic "no generator" experiment results. On the left, we show exemplar point clouds generated by different discriminators. On the right, we plot the training curves for the experiments. The $x$ - and $y$ -axis respectively represent training iterations and Wasserstein estimates.
+
+<table><tr><td>Discriminator</td><td>FPD-Mix↓</td><td>FPD-Max↓</td><td>FGD↓</td><td>MMD-E↓</td><td>MMD-C↓</td><td>COV-E↑</td><td>COV-C↑</td></tr><tr><td>PointNet-Max</td><td>3.828</td><td>2.456</td><td>19.279</td><td>0.1039</td><td>0.00287</td><td>26</td><td>56</td></tr><tr><td>PointNet-Avg</td><td>19.640</td><td>19.604</td><td>39.793</td><td>0.1154</td><td>0.00642</td><td>32</td><td>12</td></tr><tr><td>PointNet-Mix</td><td>1.835</td><td>2.208</td><td>13.197</td><td>0.0898</td><td>0.00331</td><td>50</td><td>51</td></tr><tr><td>PointNet-Attention</td><td>1.350</td><td>1.642</td><td>12.359</td><td>0.1622</td><td>0.00271</td><td>52</td><td>48</td></tr><tr><td>PointNet++</td><td>15.081</td><td>14.187</td><td>33.958</td><td>0.1155</td><td>0.00503</td><td>19</td><td>37</td></tr><tr><td>DGCNN</td><td>10.629</td><td>9.939</td><td>14.882</td><td>0.0932</td><td>0.00396</td><td>46</td><td>48</td></tr><tr><td>KPConv</td><td>41.15</td><td>49.96</td><td>57.32</td><td>0.2695</td><td>0.01135</td><td>6</td><td>2.5</td></tr><tr><td>PointConv</td><td>27.69</td><td>27.05</td><td>46.82</td><td>0.2832</td><td>0.00721</td><td>3</td><td>8</td></tr></table>
+
+Table 4: Quantitative evaluation for the diagnostic "no generator" experiments. We see that PointNet-Max/Mix/Attention are successful while PointNetAvg/PointNet++/DGCNN/KPConv/PointConv fail.
+
+clouds as the fake data. Then, we conduct adversarial training where the discriminator gradients directly update the location of each points in the learnable point clouds. We adopt the same training protocol and losses as experiments in Sec. 5.2. The goal of this "no generator" experiment is to see whether the discriminator can modify the point clouds to approximate the real shapes, or in other words, we factorize the impact from generator architectures and only focus whether the gradients from discriminators can guide shape generation. Here $M = 1 0 0$ , $\cdot$ , and $B = 3 2$ . We train each experiment for 10,000 epoches until convergence.
+
+Results and Analysis. Table 4 summarizes the results and Figure 3 (left) shows exemplar generated point clouds. We clearly see that only PointNet-Max/Mix/Attention can successfully modify the learnable point clouds to get high quality shapes, while PointNet $^ { + + }$ /DGCNN/KPConv/PointConv/PointNetAvg produce much worse results. Figure 3 (right) presents the training curves of the Wasserstein estimates, which essentially describe the gap between the real and fake scores. The large score gaps for PointNet $^ { + + }$ , DGCNN and KPConv throughout the training indicate the two models are very discriminative in telling apart the fake samples from the real data. However, their gradients simply don’t help improve the learned point clouds. Note that both of them leverage relative point positions/edge information during their feature extraction, which leads to a huge amount of gradients flowing along the surface direction focusing on changing sampling patterns instead of supervising shape geometry. As a suggestion, for developing future discriminators, people may need to avoid using the relative position features. For PointNet-Avg, the failure is simply because the discriminating power is not sufficient, evident from the low Wasserstein estimates. Note that training such "no generator" experiment shares a similar flavor to the SGD sampling in introspective CNN(Jin et al., 2017).
+
+# 6 CONCLUSION AND SUGGESTIONS FOR FUTURE DISCRIMINATOR DESIGN
+
+In this work, we study the importance of sampling for 3D point cloud GAN design. We propose the sampling spectrum of discriminators and evaluation metrics that provide insights on the behaviors of point cloud GANs. We propose several empirical experiments for identifying the sampling sensitivity of a discriminator or an evaluation metric. Experiments indicate that, no matter what generator is employed, a sampling-insensitive discriminator, e.g. the commonly used PointNet-Max, will produce point cloud shapes with non-uniform density and clustering artifacts, while a sampling-oversensitive discriminator (e.g. PointNet+ $^ +$ , DGCNN, PointConv, KPConv) will lead to disastrous failures when adapting them to training point cloud GANs. Interestingly, a simple PointNet-Mix baseline coupled with the most naive fully-connected generator achieves the best performance, indicating that the current bottleneck of point cloud GAN is on the discriminator side. For future discriminators, we suggest they should be more discriminative in shape and aware but not oversensitive to sampling patterns. For the sanity check of any novel discriminators, one can leverage our proposed "no generator" experiment.
+
+# REFERENCES
+
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+
+Angel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, et al. Shapenet: An information-rich 3d model repository. arXiv preprint arXiv:1512.03012, 2015.
+
+Haoqiang Fan, Hao Su, and Leonidas J Guibas. A point set generation network for 3d object reconstruction from a single image. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 605–613, 2017.
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+Thibault Groueix, Matthew Fisher, Vladimir G Kim, Bryan C Russell, and Mathieu Aubry. A papier-mâché approach to learning 3d surface generation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 216–224, 2018.
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+Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet $^ { + + }$ : Deep hierarchical feature learning on point sets in a metric space. In Advances in neural information processing systems, pp. 5099–5108, 2017b.
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+    {
+        "type": "text",
+        "text": "RETHINKING SAMPLING IN 3D POINT CLOUD GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            454,
+            232,
+            544,
+            247
+        ],
+        "page_idx": 0
+    },
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+        "text": "In this paper, we examine the long-neglected yet important effects of point sampling patterns in point cloud GANs. Through extensive experiments, we show that sampling-insensitive discriminators (e.g. PointNet-Max) produce shape point clouds with point clustering artifacts while sampling-oversensitive discriminators (e.g. PointNet++, DGCNN, PointConv, KPConv) fail to guide valid shape generation. We propose the concept of sampling spectrum to depict the different sampling sensitivities of discriminators. We further study how different evaluation metrics weigh the sampling pattern against the geometry and propose several perceptual metrics forming a sampling spectrum of metrics. Guided by the proposed sampling spectrum, we discover a middle-point sampling-aware baseline discriminator, PointNet-Mix, which improves all existing point cloud generators by a large margin on sampling-related metrics. We point out that, given that recent research has been focused on the generator design, the discriminator design needs more attention. Our work provides both suggestions and tools for building future discriminators. We will release the code to facilitate future research. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            503,
+            336,
+            518
+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "Point cloud, as the most common form of 3D sensor data, has been widely used in a variety of 3D vision applications due to its compact yet expressive nature and its amenability to geometric manipulations. It is natural to consider how to generate point cloud through deep learning approaches, which has been a popular research topic recently. The previous research efforts in the community have been mainly devoted to conditional generation of point clouds with 3D supervision. The condition could either be images (Fan et al., 2017; Groueix et al., 2018; Park et al., 2019) or partial point clouds (Li et al., 2019; Yang et al., 2018). ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/c2faa45a97279dcce0b62c1f7a939ffd1317175ba945bb4c981516043c089985.jpg",
+        "image_caption": [
+            "Figure 1: We visualize the behavior of different discriminators when judging four different chair point clouds. (a) and (b) are generated results, (c) and (d) are point clouds sampled using FPS and uniform sampling from a real chair surface. When training the discriminators on data like (d), different discriminators make distinct decisions on the point cloud realness forming a sampling spectrum ranging from sampling-insensitive, sampling-aware to sampling-oversensitive. We advocate samplingaware discriminators in the middle of the spectrum, which provide good guidance for fixing geometric flaws (a) and big sample artifacts (b) and tolerate subtle sampling differences between (c) and (d). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "text": "Generating 3D point clouds with GANs in an unsupervised manner is an important but less explored problem. 3D point cloud GAN learns to transform a random latent code into a 3D surface point cloud by playing an adversarial game. Its development is still in an early stage compared with 2D image GANs. While existing works such as (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019) have developed a variety of generators, they all use PointNet (Qi et al., 2017a) with max pooling (PointNet-Max) as their discriminator. PointNet, which is essentially a pointwise MLP followed by a global pooling operation, is too limited in capturing shape details for a successful GAN. However, advanced networks, e.g. PointNet++(Qi et al., 2017b), DGCNN(Wang et al., 2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019), which leverage relative positions between points and hierarchical feature extraction, may not help. From our empirical study, we find they all fail to be a functioning discriminator. Understanding their failure mechanism and improving discriminator design are hence important and urgent. ",
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+        "type": "text",
+        "text": "To design a better discriminator, we first need to answer the following question: what should the discriminator examine for improving the generation quality? Or, even more fundamentally, what does it mean by the quality of generated point clouds? ",
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+        "type": "text",
+        "text": "Since a shape point cloud are the points sampled from an object surface, its quality should be evaluated from two perspectives: the depicted surface geometry and the point sampling. Arguably, geometry plays a decisive role and should be the main focus of a discriminator. However, when the generated point clouds have a good shape, there is still a full spectrum on how much a discriminator cares about the sampling patterns. We introduce the concept of Sampling Spectrum to depict the sampling sensitivity of discriminators, as illustrated in Figure 1. A sampling-insensitive discriminator (e.g. PointNet-Max) may ignore the point density variations as long as it perceives a good overall shape. Such a discriminator could identify big geometric flaws as shown in Figure 1 (a), but turns a blind eye to highly non-uniform density distribution, e.g. point clusters in Figure 1 (b). On the other extreme, a sampling-oversensitive discriminator (e.g. PointNet $^ { + + }$ , DGCNN, KPConv) can even tell the subtle difference in sampling patterns, e.g. between furthest point sampling (FPS) in Figure 1 (c) and uniform sampling in Figure 1 (d), and hence can be very narrow-minded about what a real point cloud should look like. A sampling-aware discriminator, e.g. PointNet-Mix/Attention, which lies in the middle of the spectrum, is able to identify density-related artifacts such as point clusters in Figure 1 (b), while not being too sensitive to different sampling patterns, such as Figure 1 (c) and (d). ",
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+    {
+        "type": "text",
+        "text": "Resembling the sampling spectrum of discriminators, we also examine the existing point-cloud GAN evaluation metrics from the perspective of sampling sensitivity and propose several perceptual metrics forming a Sampling Spectrum of evaluation metrics. Understanding how the metrics weigh between sampling and geometry is a prerequisite for evaluating point cloud GANs. Many of the existing sampling-insensitive metrics only evaluate the geometry factor of the generated point clouds shapes, which are blind to the obvious point clustering artifacts and uneven point density. We propose novel sampling-sensitive metrics to further complete the spectrum of point-cloud GAN evaluation metrics. ",
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+        "type": "text",
+        "text": "Guided by the proposed sampling spectrum of discriminators and evaluation metrics, experiments show that different discriminators in the spectrum could provide very different suggestions to improve a generator according to its sampling sensitivity. Sampling-insensitive discriminators, e.g. PointNetMax, is unaware of point density variations and hence its generated point cloud inevitably suffer from clustering artifacts, while sampling-oversensitive discriminators, e.g. PointNe $^ { + + }$ , DGCNN, KPConv, PointConv, simply fail to function as the discriminators and can generate much degraded point cloud shapes. We design a diagnostic “no generator” experiment to factor out the impact from generators and reveal that the gradients of sampling-oversensitive discriminators prioritize adjusting sampling patterns over producing better shape geometry. Picking a middle-point on the sampling spectrum, we discover a simple yet effective sampling-aware discriminator, PointNet-Mix, and find that it can supervise both shape generation and point density uniformity. It improves all existing generators by a large margin on sampling-related metrics. Surprisingly, we find that even the most naive fully-connected generator, coupled with PointNet-Mix, simply beats all the start-of-the-art point cloud GANs. This discovery conveys an important message to the community: instead of focusing on the generator design, people should invest more time into discriminator and seek for more powerful sampling-aware discriminators. ",
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+    {
+        "type": "text",
+        "text": "2 POINT CLOUD GAN LANDSCAPE ",
+        "text_level": 1,
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+        "text": "In this section, we review the current state of point cloud GAN covering the generators, the discriminators, and the evaluation metrics we are examining in this work. ",
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+        "type": "text",
+        "text": "2.1 POINT CLOUD GAN GENERATORS ",
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+        "type": "text",
+        "text": "Recent point cloud GAN works primarily focus on the generator design. The generator takes input a random noise $\\cdot$ and outputs a point cloud $\\boldsymbol { p } \\in \\mathbb { R } ^ { N \\times 3 }$ . The existing generators can be categorized into two classes: fully-connected (FC) generators and graph convolutional generators. The first point cloud GAN, r-GAN (Achlioptas et al., 2018), simply uses an FC network as its generator. GraphCNN-GAN (Valsesia et al., 2018) and TreeGAN (Shu et al., 2019) are the rest two published works in this field that use graph convolutional generators. The two methods are very similar in principle. The main difference lies in how they build graphs. GraphCNN-GAN builds a dynamic $k$ -nn graph based upon feature space distance while TreeGAN enforces a tree structure throughout its sequential graph expansion and the messages can only be passed from ancestors vertices to descendants vertices. ",
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+    {
+        "type": "text",
+        "text": "Deformation-based decoders (Groueix et al., 2018; Yang et al., 2018) are widely used in the point cloud auto-encoder networks for 3D shape reconstruction. The decoders leverage Multiple-layer Perceptrons (MLP) to deform template surfaces into shape surfaces taking as inputs the concatenation of template point coordinates and the latent shape feature vectors. Though the decoders can truly act as generators for point cloud GANs, they have not yet been used in unconditioned point cloud GAN literature. Recently, Mo et al. (Mo et al., 2020) use deformation-based decoder as part generators for structure-conditioned point cloud GAN. ",
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+    {
+        "type": "text",
+        "text": "2.2 POINT CLOUD GAN DISCRIMINATORS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "All existing works on unconditioned point cloud GANs use PointNet with max-pooling (PointNetMax) as their discriminators. The discriminator takes input a point cloud $p \\in \\mathbb { R } ^ { N \\times 3 }$ and outputs a score from 0 to 1. PointNet learns a function $h$ that maps each point $p _ { i }$ in the point cloud to a per-point feature $h ( p _ { i } ) \\in \\mathbb { R } ^ { d }$ and then extracts a permutation-invariant global feature $F \\in \\mathbb { R } ^ { d }$ by pooling the per-point features across all points using a symmetric function $g$ , which can be max pooling, average pooling, etc. Namely, we have $F = \\overline { { g } } ( \\{ \\dot { h ( p _ { 1 } ) } , h ( p _ { 2 } ) , \\cdots , h ( \\overline { { p _ { N } } } ) \\} )$ . ",
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+    {
+        "type": "text",
+        "text": "PointNet-Max/Avg. Though $g$ can be any symmetric function, most existing works use maxpooling. In PointNet, the authors show that max-pooling outperforms average-pooling on 3D shape classification tasks. They further show that the global feature $F$ obtained from max-pooling is determined by only a sparse subset of the points, namely critical points $\\mathcal { C } _ { S }$ , bringing PointNet-Max with robustness against small data perturbation and corruption. However, this property may limit its discriminative power on telling the density variations and classifying different sampling patterns. To investigate how different aggregation operations affect the sampling sensitivity and generation quality, we study the two common choices of the symmetric function $g$ , max-pooling, and average-pooling, in this paper. We will show in Sec. 3 that using different aggregation operations makes huge differences when adapting PointNet-based networks as point cloud GAN discriminators. ",
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+    {
+        "type": "text",
+        "text": "PointNet-Mix. By simply concatenating the max-pooling feature and the average pool feature, we obtain another permutation-invariant feature. We name this PointNet-Mix. Formally, $F _ { \\mathrm { m i x } } =$ $[ \\operatorname* { m a x } \\{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \\}$ ; $\\arg \\{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \\} ] \\in \\mathbb { R } ^ { 2 d }$ . The mix-pooling operation is a special choice of the symmetric function $g$ . We will show in our experiments that PointNet-Mix, though simple, surprisingly improves the performance for most point cloud GANs by a large margin on sampling-aware metrics. ",
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+    {
+        "type": "text",
+        "text": "PointNet-Attention. A recent point cloud upsampling work (Li et al., 2019) incorporates a selfattention module into PointNet for its discriminator and shows improved point density in the upsampled point cloud. We denote this discriminator as PointNet-Attention. The self-attention module learns three separate MLPs to transform each $h ( p _ { i } )$ into $f ( p _ { i } ) , l ( p _ { i } ) , k ( p _ { i } ) \\in \\mathbb { R } ^ { d }$ correspondingly. Then an attention weight matrix is formed by $W =$ SoftMax $\\left( [ f ( p _ { 1 } ) , . . . , f ( p _ { N } ) ] ^ { T } [ ( l ( p _ { 1 } ) , . . . , l ( p _ { N } ) ) ] \\right) \\ \\in$ $\\mathbb { R } ^ { N \\times N }$ . A weighted features is then obtained through $w ( p _ { i } ) \\ = \\ h ( p _ { i } ) \\ + \\ W ^ { T } k ( p _ { i } )$ . The final aggregated features of PointNet-Attention is a max-pooling of the weighted feature, namely $F _ { \\mathrm { a t t e n t i o n } } = \\operatorname* { m a x } \\{ w ( p _ { 1 } ) , . . . , w ( p _ { N } ) \\}$ . Note that PointNet-Attention allows the per-point features to communicate with each other according to their similarity, which is more sensitive to sampling patterns than PointNet-Max. We investigate this strategy in our paper as well. ",
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+    {
+        "type": "text",
+        "text": "Discriminators beyond PointNet. Recently, there have been many works (Qi et al., 2017a; Li et al., 2018; Wang et al., 2019; Hermosilla et al., 2018; Thomas et al., 2019) extending PointNet to more advanced 3D deep learning architectures on point clouds. They improve PointNet by extracting more local or hierarchical geometric features via point cloud convolutions or point cloud graph learning. Though proven to be effective on shape classification and segmentation tasks, no published work examines adapting them as point cloud GAN discriminators. In this paper, we investigate four exemplar beyond-PointNet discriminators: PointNet $^ { + + }$ (Qi et al., 2017b), DGCNN (Wang et al., 2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019). ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "2.3 POINT CLOUD GAN EVALUATION METRICS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Achlioptas et al. (Achlioptas et al., 2018) introduces two distance metrics in the Euclidean space for evaluating point cloud GAN: the coverage scores (COV) computing the fraction of the point clouds in $B$ that are closed to the point clouds in $A$ using either Earth Mover’s distance (EMD) or Chamfer distance (CD), and the Minimum matching distance (MMD) scores measuring the fidelity of $A$ with respect to $B$ using either EMD or CD. In the field of 2D image GAN, it is common to use perceptual metrics, such as the Frechét distance (Heusel et al., 2017) between real and fake Gaussian measures in the feature spaces, for evaluating the generated image results. Formally, Frechet Distance $= | | \\mu _ { r } - \\mu _ { g } | | ^ { 2 } + \\operatorname { T r } ( \\Sigma _ { r } + \\Sigma _ { g } - 2 \\left( \\Sigma _ { r } \\Sigma _ { g } \\right) ^ { 1 / 2 } )$ , where $\\mu$ and $\\Sigma$ are the mean vector and the covariance matrix of the features calculated from either real or fake data distribution, and $\\mathrm { T r }$ is the matrix trace. For point clouds, Shu et al. (Shu et al., 2019) proposes Frechét Point Cloud Distance (FPD), which uses the features extracted from a pre-trained PointNet-Max model. In this paper, we position the existing metrics on a sampling spectrum according to their sampling sensitivity and propose novel sampling-aware metrics to augment the spectrum. ",
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+    {
+        "type": "text",
+        "text": "3 SAMPLING SPECTRUM OF DISCRIMINATORS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "While recent works propose many advanced generator improvements for point cloud GANs, we find that designing a good discriminator is of equal importance, if not more. In this section, we introduce the sampling spectrum of discriminators, on which we thoroughly examine the positions of different discriminators that explain their behaviors when training point cloud GANs. ",
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+        "type": "text",
+        "text": "3.1 SAMPLING SENSITIVITY OF DISCRIMINATORS ",
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+        "type": "text",
+        "text": "The sampling sensitivity of a discriminator depicts how much it responds to a change in the point density or sampling pattern of an input point cloud. We find it extremely hard to quantitatively measure this sensitivity given the difficulty of measuring changes in the sampling patterns. Naively using Euclidean metrics (e.g. CD or EMD) to measure the distance between two sampled point sets is not a solution, since given the same distance budget, the discriminator’s responses can be dramatically different depending on how the sampled points move. Instead, we can set landmarks in the continuous spectrum of sampling sensitivity by examining the discriminative power of the discriminators against different sampling patterns under a series of experiments from easy to hard. Specifically, we design two experiments to test whether a discriminator could tell clustering artifacts in point clouds and whether it could distinguish between FPS and uniform sampling patterns. Accordingly, we divide the spectrum into three regimes: sampling-insensitive/-aware/-oversensitive, as shown in Fig.1. ",
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+        "type": "text",
+        "text": "A sampling-insensitive discriminator does not respond to local point density changes if the overall shape remains roughly the same. This kind of discriminators can’t tell clustering artifacts, i.e. Fig.1 (b), and thus may cause the non-uniform density in the generated point clouds. ",
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+        "type": "text",
+        "text": "A sampling-aware discriminator can notice the significant point non-uniformity and is hence capable of supervising the generator to enforce a similar sampling distribution to the training data, while being ignorant to subtle changes when the sampling is already uniform in an intermediate scale. Such discriminator will judge Fig.1 (b) as fake but can’t tell the difference between Fig.1 (c) and (d). ",
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+    {
+        "type": "text",
+        "text": "A sampling-oversensitive discriminator can tell very subtle difference in the sampling patterns, e.g.   \nbetween FPS and uniform sampling, even if the underlying shapes are the same (Fig.1 (c) and (d)). ",
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+    {
+        "type": "text",
+        "text": "3.2 SAMPLING SENSITIVITY EXAMINATION RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We design diagnostic experiments to categorize point cloud GAN discriminators on the sampling spectrum: sampling-insensitive (PointNet-Max), sampling-aware (PointNet-Avg, PointNet-Mix, PointNet-Attention (Li et al., 2019)), and sampling-oversensitive (PointNet+ $^ { - + }$ , DGCNN, KPConv, PointConv). ",
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+    {
+        "type": "text",
+        "text": "The Discriminating Power against Clustering Artifacts. To examine whether the discriminators can tell clustering artifacts or not, we create a diagnostic classification dataset using the models from ShapeNet (Chang et al., 2015). Taking 100 chairs, we uniformly sample 2048 points from each shape, forming a set of real point clouds. To form a fake set of point clouds, for each chair, we first uniformly sample 1024 points, and then densely sample another 1024 points around a random position on the chair within a 0.1 radius. The real/fake point clouds are used as the labeled training dataset. We repeat the same process to generate a test dataset using a different set of 100 chairs. ",
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+    {
+        "type": "table",
+        "img_path": "images/3ae82dce49479ace6c8954ee595b6c8228b7e2314af5e5c672162bd1b4dfc6b9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Experiment</td><td>PN-Max</td><td>PN-Avg</td><td>PN-Mix</td><td>PN-Att</td><td>PN++</td><td>DGCNN</td><td>PointConv</td><td>KPConv</td></tr><tr><td>Clustering art.</td><td>56%/53%</td><td>98%/99%</td><td>97%/96%</td><td>94%/93%</td><td>100%/98%</td><td>100%/99%</td><td>100%/99%</td><td>100%/99%</td></tr><tr><td>FPS vs.uniform</td><td>50%/50%</td><td>50.3%/50%</td><td>51.5%/50%</td><td>50.1%/50%</td><td>100%/97%</td><td>100%/96%</td><td>100%/98%</td><td>100%/99%</td></tr><tr><td>ESS</td><td>0.03</td><td>0.49</td><td>0.46</td><td>0.43</td><td>0.95</td><td>0.95</td><td>0.97</td><td>0.99</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Evaluating the discriminating power of the discriminators against clustering artifacts and sutble change in sampling patterns along with their empirical sampling sensitivity (ESS). In each cell of the top two rows, the left shows the training accuracy while the right shows the test accuracy. ",
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+        "type": "text",
+        "text": "We supervisely train each discriminator to classify the real/fake point clouds. We train them 200 epochs until convergence. The training and test accuracies are shown in the first row of Table 1. Despite the huge density variation and the remarkable clustering artifacts, PointNet-Max is just barely better than a random guess while the rest of the discriminators are very successful in telling the fake from the real. This indicates that PointNet-Max is sampling-insensitive. Note that the discriminating power of a network towards certain artifacts is maximized under such supervised training scheme. A network will fail to identify such artifacts in an adversarial training scheme if it fails in a supervised training scheme. We will see in Sec. 5.2, point clouds generated by GAN using PointNet-Max as the discriminator indeed suffer from non-uniform density artifacts. ",
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+        "type": "text",
+        "text": "Our insight why PointNet-Avg/Mix can tell the artifacts but PointNet-Max fails is that the average pooling feature computes the center of the mass of the points in feature space and is hence aware to certain global non-uniform density distributions. PointNet-Attention leverages a learnable weighted averaging and is hence capable to identify the difference. ",
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+    {
+        "type": "text",
+        "text": "Distinguishing between FPS and Uniform Sampling. We construct another diagnostic dataset with real and fake data which are the same in their shapes but only differ in their sampling patterns. Specifically, we perform uniform sampling to generate real data and use FPS to generate fake data from 100 chairs. Figure 1 (c) and (d) illustrate the different sampling pattern outcomes. ",
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+    {
+        "type": "text",
+        "text": "We present the training and test accuracies in the second row of Table 1 and show that PointNet $^ { + + }$ , DGCNN, KPConv, and PointConv can perfectly distinguish the subtle difference in sampling patterns while the rest discriminators make no progress even on the training set. The experiment indicates that PointNet-Avg/Mix/Attention are sampling-aware while PointNet++, DGCNN, KPConv, and PointConv are sampling-oversensitive. ",
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+    {
+        "type": "text",
+        "text": "We believe, for PointNet+ $^ +$ , DGCNN, KPConv, and PointConv, their capability of distinguishing FPS from uniform sampling owes to their usage of relative point positions or edge information, which are highly sensitive to any change in sampling. We will show in Sec. 5.3 that their remarkable discriminating power on sampling patterns actually leads to the failures as functioning discriminators to train point cloud GANs. ",
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+    {
+        "type": "text",
+        "text": "Empirical Definition of Sampling Sensitivity Given the difficulty of measuring the change in sampling pattern, we would like to provide an empirical definition of sampling sensitivity of one discriminator according to its performance on the two experiments above: when the discriminator fails in telling clustering artifacts and telling the difference between FPS and uniform sampling pattern, its sampling sensitivity should be closed to 0; when the discriminator can tell clustering artifacts but can’t distinguish between FPS and uniform sampling, we want to assign it a sampling sensitivity closed to 0.5 ; when the discriminator can do both, we want to assign a sampling sensitivity closed to 1 to it. Based on this, we define empirical sampling sensitivity $\\cdot$ as: ",
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+    {
+        "type": "text",
+        "text": "ESS = Acc(Tell clustering art.) $^ +$ Acc(Tell FPS from uniform sampling) $^ { - 1 }$ , where Acc denotes the test accuracy of an experiment. See Table 1 for results. Using this metric, we would form a continuous sampling sensitivity spectrum. ",
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+    {
+        "type": "text",
+        "text": "4 SAMPLING SPECTRUM OF EVALUATION METRICS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Similar to the discriminator design, it is very important to understand how different evaluation metrics weigh the differences in sampling patterns against geometry quality. Thus, we introduce the sampling spectrum of evaluation metrics, which exactly resembles the sampling spectrum of discriminators introduced in Sec. 3. On the spectrum, we have sampling-insensitive metrics that measure only the shape geometry and are ignorant of the sampling patterns, and sampling-sensitive metrics that measure both at the same time. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For perceptual metrics, i.e. Frechét distances in feature spaces, the sampling sensitivity of the metric purely depends on the sampling sensitivity of its feature extractor. Frechét distance measured in different feature spaces may respond very differently to changes in point density and sampling patterns. In this work, we examine three Frechét distance metrics that extract features using PointNetMax, PointNet-Mix, and DGCNN, respectively. We denote them as Frechét PointNet-Max Distance (FPD-Max), Frechét PointNet-Mix Distance (FPD-Mix), and Frechét DGCNN Distance (FGD). We pretrain all the three feature extraction networks on ModelNet40 (Wu et al., 2015) shape classification. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2ae5f2cf923fea9d318098171de62c1544d26af427878d0d5240f4db7813e788.jpg",
+        "table_caption": [
+            "Table 2: Examining sampling sensitivity of evaluation metrics. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Data</td><td>FPD-Mix↓</td><td>FPD-Max↓</td><td>FGD↓</td><td>MMD-E↓</td><td>MMD-C↓</td><td>COV-E↑</td><td>COV-C↑</td></tr><tr><td>Uniformre-sampling</td><td>0.1153</td><td>0.0926</td><td>0.8141</td><td>0.1104</td><td>0.00145</td><td>70.69</td><td>72.16</td></tr><tr><td>Farthest point sampling</td><td>0.1700</td><td>0.1558</td><td>1.8833</td><td>0.1064</td><td>0.00137</td><td>67.74</td><td>69.36</td></tr><tr><td>Biased sampling</td><td>2.8631</td><td>0.3524</td><td>9.6719</td><td>0.2469</td><td>0.00145</td><td>23.12</td><td>71.28</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "To examine the sampling sensitivity of FPD-Mix/Max and FGD, we create several copies of the training split of our ShapeNet chair dataset (see Sec.5.1), each of which uses a different sampling strategy to obtain the shape point clouds. The reference one used as the ground truth is using uniform sampling. Then we consider 1) uniform sampling with a different random seed; 2) FPS; and 3) biased sampling with clustered artifacts (as described in Sec. 3.2). We use all the available metrics to evaluate their distances to the ground truth data. The results are shown in Table 2. ",
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+    },
+    {
+        "type": "text",
+        "text": "We observe that the Frechét distance metrics share the same sampling sensitivity of their corresponding discriminators. For example, since PointNet-Max is sampling-insensitive, FPD-Max remains very low even on biased sampling data, hence FPD-Max serves as a perceptual geometry metric, which is ignorant to sampling patterns. Similarly, we find that FPD-Mix is sampling-aware since it clearly detects the biased sampling patterns while not being able to distinguish the uniform sampling and FPS, while FGD is sampling-oversensitive in that it can tell apart FPS and uniform sampling. ",
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+    },
+    {
+        "type": "text",
+        "text": "For Euclidean distance metrics, results in Table 2 show that COV-EMD and MMD-EMD are samplingaware, which is intuitively reasonable since EMD enforces a one-to-one matching and is aware of the point density, while COV-CD and MMD-CD are sampling-insensitive. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Aware of both the sampling spectrums, we conduct experiments to further evaluate the performance of point cloud GANs under various evaluation metrics. We show that the point cloud GANs using sampling-insensitive discriminators may produce point clustering artifacts, while samplingoversensitive discriminators fail to supervise point cloud GAN training at all. We further devise a diagnostic \"no-generator\" experiment that factors out the generators to better illustrate our discoveries on discriminators. More interesting, we find that the simple PointNet-Mix paired of any generator, even with the most naive fully-connected one, achieves the state-of-the-art performance. ",
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+    {
+        "type": "text",
+        "text": "5.1 SETTING AND DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We provide a thorough comparison of all the discriminators investigated in Sec.3 combining with all the available generators in the published literature, including the FC generator proposed in rGAN (Achlioptas et al., 2018), and graph convolutional generators used in TreeGAN (Shu et al., 2019). We also add a deformation-based generator into the comparison given its popularity for supervised point cloud reconstruction (Groueix et al., 2018; Yang et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "We use two datasets to evaluate the GANs. One is a single-category dataset containing point clouds sampled from all 6,778 chair meshes in ShapeNet (Chang et al., 2015). The other is a multi-category dataset combining shapes from airplane, car, chair, rifle, sofa, table, vessel categories in ShapeNet. The multi-category dataset contains 34,313 shapes in total. We uniformly sample 2048 points from each shape to form the two datasets. We follow the $8 5 \\% / 5 \\% / 1 0 \\%$ train/validation/test split in (Achlioptas et al., 2018) and use WGAN-gp (Gulrajani et al., 2017) for the GAN training, similar to previous works (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "5.2 EVALUATING POINTNET-BASED DISCRIMINATORS WITH VARIOUS GENERATORS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We report the performance for point cloud GANs that combine PointNet-Max/Min/Attention discriminators and FC/Deform/TreeGAN/Graph-CNN generators in Table 3. We observe that GANs using PointNet-Mix as the discriminator outperform the ones using PointNet-Max/Attention across all different generators. On sampling-aware/sensitive metrics (i.e. FPD-Mix, FGD, MMD-EMD, COV-EMD), PointNet-Mix is always significantly better than PointNet-Max and is better than PointNet-Attention mainly on FPD-Mix and FGD. Regarding geometry quality evaluated using sampling-insensitive metrics (i.e. FPD-Max, MMD-CD, COV-CD), PointNet-Mix is always significantly better than PointNet-Attention on FPD-Max and COV-CD while remaining on a par with PointNet-Max. ",
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+        "img_path": "images/bb0b6d2514c2aef60eea70ee2a670c45a56e495ce19debe24e184b17ef906f14.jpg",
+        "table_caption": [
+            "Table 3: Evaluating PointNet-based discriminators with different generators. We observe that PointNet-Mix can significantly improve all sampling-aware/sensitive metric, including FPD-Mix, FGD, MMD-EMD, COV-EMD for all the generators. When paired with FC generator, PointNet-Mix achieves the best performance. We include PointFlow(Yang et al., 2019) and Oracle for a comparison. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Generator</td><td rowspan=1 colspan=1>Pooling</td><td rowspan=1 colspan=1>FPD-Mix↓</td><td rowspan=1 colspan=1>FPD-Max↓</td><td rowspan=1 colspan=1>FGD↓</td><td rowspan=1 colspan=1>MMD-E↓</td><td rowspan=1 colspan=1>MMD-C↓</td><td rowspan=1 colspan=1>COV-E↑</td><td rowspan=1 colspan=1>COV-C↑</td></tr><tr><td rowspan=10 colspan=1>Chair</td><td rowspan=2 colspan=1>FCFC</td><td rowspan=2 colspan=1>MaxMix</td><td rowspan=2 colspan=1>1.5710.184</td><td rowspan=2 colspan=1>0.2110.209</td><td rowspan=1 colspan=1>7.030</td><td rowspan=1 colspan=1>0.1017</td><td rowspan=1 colspan=1>0.00164</td><td rowspan=1 colspan=1>23.56</td><td rowspan=1 colspan=1>72.75</td></tr><tr><td rowspan=1 colspan=1>2.124</td><td rowspan=1 colspan=1>0.0674</td><td rowspan=1 colspan=1>0.00196</td><td rowspan=1 colspan=1>73.64</td><td rowspan=1 colspan=1>74.96</td></tr><tr><td rowspan=7 colspan=1>FCDeformDeformDeformTreeGANTreeGANGraph-CNN</td><td rowspan=1 colspan=1>Attention</td><td rowspan=1 colspan=1>0.635</td><td rowspan=1 colspan=1>0.672</td><td rowspan=1 colspan=1>4.971</td><td rowspan=1 colspan=1>0.1156</td><td rowspan=1 colspan=1>0.00160</td><td rowspan=1 colspan=1>68.92</td><td rowspan=1 colspan=1>70.54</td></tr><tr><td rowspan=2 colspan=1>MaxMix</td><td rowspan=2 colspan=1>0.9130.534</td><td rowspan=2 colspan=1>0.2680.373</td><td rowspan=1 colspan=1>5.602</td><td rowspan=2 colspan=1>0.09080.0695</td><td rowspan=2 colspan=1>0.002010.00200</td><td rowspan=2 colspan=1>68.576.29</td><td rowspan=2 colspan=1>72.6175.11</td></tr><tr><td rowspan=1 colspan=1>2.836</td></tr><tr><td rowspan=4 colspan=1>AttentionMaxMixMax</td><td rowspan=1 colspan=1>0.696</td><td rowspan=1 colspan=1>0.755</td><td rowspan=1 colspan=1>2.987</td><td rowspan=1 colspan=1>0.1141</td><td rowspan=1 colspan=1>0.00160</td><td rowspan=1 colspan=1>68.77</td><td rowspan=1 colspan=1>69.36</td></tr><tr><td rowspan=3 colspan=1>1.4420.2931.034</td><td rowspan=1 colspan=1>0.654</td><td rowspan=2 colspan=1>7.8084.032</td><td rowspan=2 colspan=1>0.09620.0704</td><td rowspan=2 colspan=1>0.001910.00211</td><td rowspan=3 colspan=1>24,7474.8248.90</td><td rowspan=3 colspan=1>73.4978.7963.18</td></tr><tr><td rowspan=1 colspan=1>0.334</td></tr><tr><td rowspan=1 colspan=1>0.981</td><td rowspan=1 colspan=1>7.494</td><td rowspan=1 colspan=1>0.0812</td><td rowspan=1 colspan=1>0.00191</td></tr><tr><td rowspan=1 colspan=2>PointFlowOracle</td><td rowspan=1 colspan=1>1.7820.088</td><td rowspan=1 colspan=1>1.4050.093</td><td rowspan=1 colspan=1>3.2560.814</td><td rowspan=1 colspan=1>0.13350.0594</td><td rowspan=1 colspan=1>0.001900.00165</td><td rowspan=1 colspan=1>71.7679.56</td><td rowspan=1 colspan=1>71.7379.69</td></tr><tr><td rowspan=9 colspan=1>Multi-Cat</td><td rowspan=3 colspan=1>FCFCFC</td><td rowspan=2 colspan=1>MaxMix</td><td rowspan=1 colspan=1>1.553</td><td rowspan=1 colspan=1>0.354</td><td rowspan=1 colspan=1>6.981</td><td rowspan=1 colspan=1>0.0842</td><td rowspan=1 colspan=1>0.00153</td><td rowspan=1 colspan=1>35.16</td><td rowspan=1 colspan=1>64.16</td></tr><tr><td rowspan=1 colspan=1>0.255</td><td rowspan=1 colspan=1>0.285</td><td rowspan=1 colspan=1>2.550</td><td rowspan=1 colspan=1>0.0656</td><td rowspan=2 colspan=1>0.001840.00134</td><td rowspan=2 colspan=1>73.572.19</td><td rowspan=2 colspan=1>72.1669.63</td></tr><tr><td rowspan=1 colspan=1>Attention</td><td rowspan=1 colspan=1>0.414</td><td rowspan=1 colspan=1>0.453</td><td rowspan=1 colspan=1>4.234</td><td rowspan=1 colspan=1>0.1188</td></tr><tr><td rowspan=5 colspan=1>DeformDeformDeformTreeGANTreeGAN</td><td rowspan=2 colspan=1>MaxMix</td><td rowspan=1 colspan=1>1.072</td><td rowspan=1 colspan=1>0.633</td><td rowspan=1 colspan=1>3.845</td><td rowspan=1 colspan=1>0.0799</td><td rowspan=1 colspan=1>0.00179</td><td rowspan=1 colspan=1>62.5</td><td rowspan=1 colspan=1>64.5</td></tr><tr><td rowspan=1 colspan=1>0.614</td><td rowspan=1 colspan=1>0.349</td><td rowspan=1 colspan=1>2.451</td><td rowspan=1 colspan=1>0.0670</td><td rowspan=1 colspan=1>0.00191</td><td rowspan=1 colspan=1>70.83</td><td rowspan=2 colspan=1>68.8369.60</td></tr><tr><td rowspan=1 colspan=1>Attention</td><td rowspan=1 colspan=1>0.616</td><td rowspan=1 colspan=1>0.720</td><td rowspan=1 colspan=1>2.531</td><td rowspan=1 colspan=1>0.1113</td><td rowspan=1 colspan=1>0.00141</td><td rowspan=1 colspan=1>72.04</td></tr><tr><td rowspan=2 colspan=1>MaxMix</td><td rowspan=2 colspan=1>1.7140.388</td><td rowspan=1 colspan=1>0.437</td><td rowspan=1 colspan=1>6.342</td><td rowspan=1 colspan=1>0.1093</td><td rowspan=1 colspan=1>0.00158</td><td rowspan=2 colspan=1>25.3372.66</td><td rowspan=2 colspan=1>67.071.0</td></tr><tr><td rowspan=1 colspan=1>0.420</td><td rowspan=1 colspan=1>4.300</td><td rowspan=1 colspan=1>0.0699</td><td rowspan=1 colspan=1>0.00184</td></tr><tr><td rowspan=1 colspan=2>Oracle</td><td rowspan=1 colspan=1>0.131</td><td rowspan=1 colspan=1>0.177</td><td rowspan=1 colspan=1>0.673</td><td rowspan=1 colspan=1>0.06012</td><td rowspan=1 colspan=1>0.00124</td><td rowspan=1 colspan=1>77.14</td><td rowspan=1 colspan=1>77.57</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "In Figure 2, we present the generated point clouds for all the experiments with color-coding for the local point density. We see that the generators trained using PointNet-Max usually suffer from non-uniform density, except for the deformation generator. On the chair class, points are usually clustering around the seat area while being sparse at the back. On the contrary, PointNet-Mix enforces a globally uniform point density and hence greatly improves the visual quality of the generated point clouds. PointNet-Attention is in the between. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With PointNet-Mix, we observe that the most naive FC generator works the best outperforming the previously state-of-the-state generators on almost of metrics. This showcases the importance of being sampling-aware but not sampling-insensitive/oversensitive as a discriminator. It also suggests that future works may focus on designing more powerful sampling-aware discriminators. ",
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+    {
+        "type": "text",
+        "text": "5.3 DIAGNOSING FAILURES FOR VARIOUS DISCRIMINATORS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Observing that the discriminator forms the bottleneck of the point cloud GAN, it is important to examine whether advanced networks, e.g. PointNet+ $^ +$ (Qi et al., 2017b), DGCNN (Wang et al., 2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019) can outperform PointNet-based discriminator. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Surprisingly, we fail to observe any of them ever to be successful to train a GAN in our extensive experiments, in which we perform a systematic search, including varying generator architecture, changing hyperparameters (e.g. the number of layers and parameters in discriminator, learning rate), whether use batch norm, etc. During GAN training, we observe the four advanced discriminator behave very similarly: they are very discriminative in the sense that the gaps between the real and fake scores remain significantly large, but the quality of the generated point clouds stays very bad. The contradictory behaviors seem unreasonable at the first glance: how can a discriminator be very discriminative but teach nothing to the generator? Thus, we design a diagnostic \"no generator\" experiment to examine the underlying reasons. ",
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+    {
+        "type": "text",
+        "text": "No Generator Experiments. During GAN training, the gradients from discriminator output scores back-propagate to the generated point clouds and then further back-propagate to the weights of the generator to update the generator. We design a diagnostic experiment that removes the generator and examines whether the discriminator gradients are informative enough for supervising point cloud GANs. ",
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+    {
+        "type": "text",
+        "text": "Concretely, we take $\\cdot$ real chairs as real data and we randomly initialize $M$ learnable point clouds $p \\in \\mathbb { R } ^ { N \\times 3 }$ with i.i.d Gaussian noises $\\cdot$ . For each training iteration, we sample $\\cdot$ point clouds from the real dataset as the real data and sample $\\cdot$ point clouds from the learnable point ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9ab46f0bd251aa0ef4577907186cc5ad8e8d73246239da39a1cce07f60003cd8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/e359eb374e57942b0203cde57d30eb2a3b5555e59ff8408d26bfd94052794c2d.jpg",
+        "image_caption": [
+            "Figure 2: Visualization of point clouds generated by different methods. We show generated point clouds in few exemplar shapes for a fair and easier comparison regarding their sampling quality and geometry quality. We color-code the local point density that ranges from sparse (dark blue) to dense (light yellow). ",
+            "Figure 3: The diagnostic \"no generator\" experiment results. On the left, we show exemplar point clouds generated by different discriminators. On the right, we plot the training curves for the experiments. The $x$ - and $y$ -axis respectively represent training iterations and Wasserstein estimates. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/a6502f96127bea14acbd72cf43811ba4be0dad6184ce827043f8ab3e9573dc5e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Discriminator</td><td>FPD-Mix↓</td><td>FPD-Max↓</td><td>FGD↓</td><td>MMD-E↓</td><td>MMD-C↓</td><td>COV-E↑</td><td>COV-C↑</td></tr><tr><td>PointNet-Max</td><td>3.828</td><td>2.456</td><td>19.279</td><td>0.1039</td><td>0.00287</td><td>26</td><td>56</td></tr><tr><td>PointNet-Avg</td><td>19.640</td><td>19.604</td><td>39.793</td><td>0.1154</td><td>0.00642</td><td>32</td><td>12</td></tr><tr><td>PointNet-Mix</td><td>1.835</td><td>2.208</td><td>13.197</td><td>0.0898</td><td>0.00331</td><td>50</td><td>51</td></tr><tr><td>PointNet-Attention</td><td>1.350</td><td>1.642</td><td>12.359</td><td>0.1622</td><td>0.00271</td><td>52</td><td>48</td></tr><tr><td>PointNet++</td><td>15.081</td><td>14.187</td><td>33.958</td><td>0.1155</td><td>0.00503</td><td>19</td><td>37</td></tr><tr><td>DGCNN</td><td>10.629</td><td>9.939</td><td>14.882</td><td>0.0932</td><td>0.00396</td><td>46</td><td>48</td></tr><tr><td>KPConv</td><td>41.15</td><td>49.96</td><td>57.32</td><td>0.2695</td><td>0.01135</td><td>6</td><td>2.5</td></tr><tr><td>PointConv</td><td>27.69</td><td>27.05</td><td>46.82</td><td>0.2832</td><td>0.00721</td><td>3</td><td>8</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Quantitative evaluation for the diagnostic \"no generator\" experiments. We see that PointNet-Max/Mix/Attention are successful while PointNetAvg/PointNet++/DGCNN/KPConv/PointConv fail. ",
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+    },
+    {
+        "type": "text",
+        "text": "clouds as the fake data. Then, we conduct adversarial training where the discriminator gradients directly update the location of each points in the learnable point clouds. We adopt the same training protocol and losses as experiments in Sec. 5.2. The goal of this \"no generator\" experiment is to see whether the discriminator can modify the point clouds to approximate the real shapes, or in other words, we factorize the impact from generator architectures and only focus whether the gradients from discriminators can guide shape generation. Here $M = 1 0 0$ , $\\cdot$ , and $B = 3 2$ . We train each experiment for 10,000 epoches until convergence. ",
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+    {
+        "type": "text",
+        "text": "Results and Analysis. Table 4 summarizes the results and Figure 3 (left) shows exemplar generated point clouds. We clearly see that only PointNet-Max/Mix/Attention can successfully modify the learnable point clouds to get high quality shapes, while PointNet $^ { + + }$ /DGCNN/KPConv/PointConv/PointNetAvg produce much worse results. Figure 3 (right) presents the training curves of the Wasserstein estimates, which essentially describe the gap between the real and fake scores. The large score gaps for PointNet $^ { + + }$ , DGCNN and KPConv throughout the training indicate the two models are very discriminative in telling apart the fake samples from the real data. However, their gradients simply don’t help improve the learned point clouds. Note that both of them leverage relative point positions/edge information during their feature extraction, which leads to a huge amount of gradients flowing along the surface direction focusing on changing sampling patterns instead of supervising shape geometry. As a suggestion, for developing future discriminators, people may need to avoid using the relative position features. For PointNet-Avg, the failure is simply because the discriminating power is not sufficient, evident from the low Wasserstein estimates. Note that training such \"no generator\" experiment shares a similar flavor to the SGD sampling in introspective CNN(Jin et al., 2017). ",
+        "bbox": [
+            173,
+            138,
+            825,
+            337
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND SUGGESTIONS FOR FUTURE DISCRIMINATOR DESIGN ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we study the importance of sampling for 3D point cloud GAN design. We propose the sampling spectrum of discriminators and evaluation metrics that provide insights on the behaviors of point cloud GANs. We propose several empirical experiments for identifying the sampling sensitivity of a discriminator or an evaluation metric. Experiments indicate that, no matter what generator is employed, a sampling-insensitive discriminator, e.g. the commonly used PointNet-Max, will produce point cloud shapes with non-uniform density and clustering artifacts, while a sampling-oversensitive discriminator (e.g. PointNet+ $^ +$ , DGCNN, PointConv, KPConv) will lead to disastrous failures when adapting them to training point cloud GANs. Interestingly, a simple PointNet-Mix baseline coupled with the most naive fully-connected generator achieves the best performance, indicating that the current bottleneck of point cloud GAN is on the discriminator side. For future discriminators, we suggest they should be more discriminative in shape and aware but not oversensitive to sampling patterns. For the sanity check of any novel discriminators, one can leverage our proposed \"no generator\" experiment. ",
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parse/train/ALvt7nXa2q/ALvt7nXa2q.md

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+# Overcoming Catastrophic Forgetting in Incremental Few-Shot Learning by Finding Flat Minima
+
+Guangyuan Shi,∗ Jiaxin Chen∗, Wenlong Zhang, Li-Ming Zhan, Xiao-Ming Wu† Department of Computing The Hong Kong Polytechnic University {guang-yuan.shi, jiax.chen, wenlong.zhang, lmzhan.zhan}@connect.polyu.hk xiao-ming.wu@polyu.edu.hk
+
+# Abstract
+
+This paper considers incremental few-shot learning, which requires a model to continually recognize new categories with only a few examples provided. Our study shows that existing methods severely suffer from catastrophic forgetting, a well-known problem in incremental learning, which is aggravated due to data scarcity and imbalance in the few-shot setting. Our analysis further suggests that to prevent catastrophic forgetting, actions need to be taken in the primitive stage – the training of base classes instead of later few-shot learning sessions. Therefore, we propose to search for flat local minima of the base training objective function and then fine-tune the model parameters within the flat region on new tasks. In this way, the model can efficiently learn new classes while preserving the old ones. Comprehensive experimental results demonstrate that our approach outperforms all prior state-of-the-art methods and is very close to the approximate upper bound. The source code is available at https://github.com/moukamisama/F2M.
+
+# 1 Introduction
+
+Why study incremental few-shot learning? Incremental learning enables a model to continually learn new concepts from new data without forgetting previously learned knowledge. Rooted from real-world applications, this topic has attracted a significant amount of interest in recent years [5, 30, 31, 40, 26]. Incremental learning assumes sufficient training data is provided for new classes, which is impractical in many application scenarios, especially when the new classes are rare categories which are costly or difficult to collect. This motivates the study of incremental few-shot learning, a more difficult paradigm that aims to continually learn new tasks with only a few examples.
+
+Challenges. The major challenge for incremental learning is catastrophic forgetting [14, 28, 35], which refers to the drastic performance drop on previous tasks after learning new tasks. This phenomenon is caused by the inaccessibility to previous data while learning on new data. Catastrophic forgetting presents a bigger challenge for incremental few-shot learning. Due to the small amount of training data in new tasks, the model tends to severely overfit on new classes while quickly forgetting old classes, resulting in catastrophic performance.
+
+Current research. The study of incremental few-shot learning has just started [47, 41, 60, 9, 8, 34, 59]. Current works mainly borrow ideas from research in incremental learning to overcome the forgetting problem, by enforcing strong constraints on model parameters to penalize the changes of parameters [34, 28, 56], or by saving a small amount of exemplars from old classes and adding constraints on the exemplars to avoid forgetting [40, 20, 4]. However, in our empirical study, we find that an intransigent model that only trains on base classes and does not tune on new tasks consistently outperforms state-of-the-art methods, including a joint-training method [47] that uses all encountered data for training and hence suffers from severe data imbalance. This observation motivates us to address this harsh problem from a different angle.
+
+Our solution. Unlike existing solutions that try to overcome the catastrophic forgetting problem during the process of learning new tasks, we adopt a different approach by considering this issue during the training of base classes. Specifically, we propose to search for flat local minima of the base training objective function. For any parameter vector in the flat region around the minima, the loss is small, and the base classes are supposed to be well separated. The flat local minima can be found by adding random noise to the model parameters for multiple times and jointly optimizing multiple loss functions. During the following incremental few-shot learning stage, we fine-tune the model parameters within the flat region, which can be achieved by clamping the parameters after updating them on few-shot tasks. In this way, the model can efficiently learn new classes while preserving the old ones. Our key contributions are summarized as follows:
+
+• We conduct a comprehensive empirical study on existing incremental few-shot learning methods and discover that a simple baseline model that only trains on base classes outperforms state-of-the-art methods, which demonstrates the severity of catastrophic forgetting. • We propose a novel approach for incremental few-shot learning by addressing the catastrophic forgetting problem in the primitive stage. Through finding the flat minima region during training on base classes and fine-tuning within the region while learning on new tasks, our model can overcome catastrophic forgetting and avoid overfitting. • Comprehensive experimental results on CIFAR-100, miniImageNet, and CUB-200-2011 show that our approach outperforms all state-of-the-art methods and achieves performance that is very close to the approximate upper bound.
+
+# 2 Related Work
+
+Few-shot learning aims to learn to generalize to new categories with a few labeled samples in each class. Current few-shot methods mainly include optimization-based methods [12, 23, 32, 39, 45, 46, 55] and metric-based methods [13, 19, 44, 49, 52, 58, 57, 53]. Optimization-based methods can achieve fast adaptation to new tasks with limited samples by learning a specific optimization algorithm. Metric-based approaches exploit different distance metrics such as L2 distance [44], cosine similarity [49], and DeepEMD [58] in the learned metric/embedding space to measure the similarity between samples. Recently, Tian et al. [48] find that standard supervised training can learn a good metric space for unseen classes, which echoes with our observation on the proposed baseline model in Sec. 3.
+
+Incremental learning focuses on the challenging problem of continually learning to recognize new classes in new coming data without forgetting old classes [6, 7, 10, 51]. Previous research mainly includes multi-class incremental learning [4, 38, 22, 33, 54, 51] and multi-task incremental learning [21, 31, 42]. To overcome the catastrophic forgetting problem, some attempts propose to impose strong constraints on model parameters by penalizing the changes of parameters [28, 1]. Other attempts try to enforce constraints on the exemplars of old classes by restricting the output logits [40] or penalizing the changes of embedding angles [20]. In this work, our empirical study shows that imposing strong constraints on the arriving new classes may not be a promising way to tackle incremental few-shot learning, due to the scarcity of training data for new classes.
+
+Incremental few-shot learning [47, 41, 60, 9, 8] aims to incrementally learn from very few samples. TOPCI [47] proposes a neural gas network to learn and preserve the topology of the feature manifold formed by different classes. FSLL [34] only selects few model parameters for incremental learning and ensures the parameters are close to the optimal ones. To overcome catastrophic forgetting, IDLVQC [8] imposes constraints on the saved exemplars of each class by restricting the embedding drift, and Zhang et al. [59] propose to fix the embedding network for incremental learning. Similar to the finding of Zhang et al., we also discover that an intransigent model that simply does not adapt to new tasks can outperform prior state-of-the-art methods.
+
+Robust optimization. It has been found that flat local minima leads to better generalization capabilities than sharp minima in the sense that a flat minimizer is more robust when the test loss is shifted due to random perturbations [18, 17, 24]. A substantial body of methods [2, 37, 11, 15] have been proposed to optimize neural networks towards flat local minima. In this paper, we show that for incremental few-shot learning, finding flat minima in the base session and tuning the model within the flat region on new tasks can significantly mitigate catastrophic forgetting.
+
+# 3 Severity of Catastrophic Forgetting in Incremental Few-Shot Learning
+
+# 3.1 Problem Statement
+
+Incremental few-shot learning (IFL) aims to continually learn to recognize new classes with only few examples. Similar to incremental learning $\left( \operatorname { I L } \right)$ , an IFL model is trained by a sequence of training sessions $\{ \mathcal { D } ^ { 1 } , \cdots , \mathcal { D } ^ { t } \}$ , where $\mathcal { D } ^ { t } = \{ z _ { i } = ( x _ { i } ^ { t } , y _ { i } ^ { t } ) \} _ { i }$ is the training data of session $t$ and $\ v x _ { i } ^ { t }$ is an example of class $y _ { i } ^ { t } \in \mathcal { C } ^ { t }$ (the class set of session $t$ ). In IFL, the base session $\mathcal { D } ^ { 1 }$ usually contains a large number of classes with sufficient training data for each class, while the following sessions $\left( t \geq 2 \right)$ ) only have a small number of classes with few training samples per class, e.g., $\bar { \mathcal { D } } ^ { t }$ is often presented as an $N$ -way $K$ -shot task with small $N$ and $K$ . The key difference between $\mathrm { I L }$ and IFL is, for $\mathrm { I L }$ , sufficient training data is provided in each session. Similar to $\mathrm { I L }$ , in each training session $t$ of IFL, the model has only access to the training data $\mathcal { D } ^ { t }$ and possibly a small amount of saved exemplars from previous sessions. When the training of session $t$ is completed, the model is evaluated on test samples from all encountered classes ${ \mathcal { C } } = \textstyle \bigcup _ { i = 1 } ^ { t } { \mathcal { C } } ^ { i }$ , where it is assumed that there is no overlap between the classes of different sessions, i.e., $\forall i , j$ and $i \neq j$ , ${ \mathcal { C } } ^ { i } \cap { \mathcal { C } } ^ { j } = \emptyset$ .
+
+Catastrophic forgetting. IFL is undoubtedly a more challenging problem than $\mathrm { I L }$ due to the data scarcity setting. IL suffers from catastrophic forgetting, a well-known phenomenon and long-standing issue, which refers to the drastic drop in test performance on previous (old) classes, caused by the inaccessibility of old data in the current training session. Unfortunately, catastrophic forgetting is an even bigger issue for IFL, because data scarcity makes it difficult to adapt well to new tasks and learn new concepts, while the adaptation process could easily lead to the forgetting of base classes. In the following, we illustrate this point by evaluating a simple baseline model for IFL.
+
+# 3.2 A Simple Baseline Model for IFL
+
+We consider an intransigent model that simply does not adapt to new tasks. Particularly, the model only needs to be trained in the base session $\bar { \mathcal { D } } ^ { 1 }$ and is directly used for inference in all sessions.
+
+Training $( t = 1$ ). We train a feature extractor $f$ parameterized by $\phi$ with a fully-connected layer as classifier by minimizing the standard cross-entropy loss using the training examples of $\mathcal { D } ^ { 1 }$ . The feature extractor $f$ is fixed for the following sessions $\left( t \geq 2 \right)$ ) without any fine-tuning on new classes.
+
+Inference (test). In each session, the inference is conducted by a simple nearest class mean (NCM) classification algorithm [36]. Specifically, all the training and test samples are mapped to the embedding space of the feature extractor $f$ , and Euclidean distance $d ( \cdot , \cdot )$ is used to measure the similarity between them. The classifier is given by
+
+$$
+c _ { k } ^ { \star } = \operatorname * { a r g m i n } _ { c \in \mathcal { C } } d ( f ( x ; \phi ) , p _ { c } ) , \mathrm { ~ w h e r e ~ } p _ { c } = \frac { 1 } { N _ { c } } \sum _ { i } \mathbb { 1 } ( y _ { i } = c ) f ( x _ { i } ; \phi ) ,
+$$
+
+where $\mathcal { C }$ denotes all the encountered classes, $p _ { c }$ refers to the prototype of class $c$ (the mean vector of all the training samples of class $c$ in the embedding space), and $N _ { c }$ denotes the number of the training images of class $c$ . Note that we save the prototypes of all classes in ${ \mathcal { C } } ^ { t }$ for later evaluation.
+
+The baseline model outperforms state-of-the-art IFL and IL methods. We compare the above baseline model against state-of-the-art IFL methods including FSLL [34], IDLVQC [8] and TOPIC [47], IL methods including Rebalance [20] and iCarl [40], and a joint-training method that uses all previously seen data including the base and the following few-shot tasks for training, for IFL. The performance is evaluated on miniImageNet, CIFAR-100, and CUB-200. We tune the methods re-implemented by us to the best performance. For the other methods, we use the results reported in the original papers. The experimental details are provided in Sec. 5. As shown in Fig. 1, the baseline model consistently outperforms all the compared methods including the joint-training method (which suffers from severe data imbalance) on every dataset3. The fact that an intransigent model performs best suggests that
+
+![](images/a3479acb9b05770da7d6c9d886cf4da1b35bd210974b3e33e48344c785552875.jpg)  
+Figure 1: Comparison of the proposed baseline model with state-of-the-art IFL and $\mathrm { I L }$ methods and the joint-training method.The baseline model outperforms all the other methods.
+
+• For IFL, preserving the old (base classes) may be more critical than adapting to the new. Due to data scarcity, the performance gain on new classes is limited and cannot make up for the significant performance drop on base classes. • Prior works [47, 8, 34, 20, 40] that enforce strong constraints on model parameters or exemplars during fine-tuning on new classes cannot effectively prevent catastrophic forgetting in IFL, indicating that actions may need to be taken in the base training stage.
+
+# 4 Overcoming Catastrophic Forgetting in IFL by Finding Flat Minima
+
+The goal of IFL is to preserve the old while adapting to the new efficiently. The results and analysis in Sec. 3 suggest that it might be “a bit late” to try to prevent catastrophic forgetting in the few-shot learning sessions $\left( t \geq 2 \right)$ ), which motivates us to consider this problem in the base training session.
+
+Overview of our approach. To overcome catastrophic forgetting in IFL, we propose to find a $b$ -flat $( b > 0$ ) local minima $\theta ^ { \star }$ of the base training objective function and then fine-tune the model within the flat region in later few-shot learning sessions. Specifically, for any parameter vector $\theta$ in the flat region, i.e., $\theta ^ { \star } - b \preceq \theta \preceq \theta ^ { \star } + b$ , the risk (loss) of the base classes is minimized such that the classes are well separated in the embedding space of $f _ { \theta }$ . In the later incremental few-shot learning sessions $\left( t \geq 2 \right)$ ), we fine-tune the model parameters within this region to learn new classes, i.e., to find
+
+$$
+\theta ^ { \prime } = \arg \operatorname* { m i n } _ { \theta } \sum _ { z \in \mathcal { D } ^ { t } } \mathcal { L } ( z ; \theta ) , \mathrm { s . t . } \theta ^ { \star } - b \preceq \theta \preceq \theta ^ { \star } + b .
+$$
+
+As such, the fine-tuned model $\theta ^ { \prime }$ can adapt to new classes while preserving the old ones. Also, due to the nature of few-shot learning, to avoid excessive training and overfitting, it suffices to tune the model in a relatively small region. A graphical illustration of our approach and prior arts, as well as the notions of sharp minima and flat minima, are presented in Fig. 2.
+
+# 4.1 Searching for Flat Local Minima in the Base Training Stage
+
+A formal definition of $b$ -flat local minima is given as follows.
+
+Definition 1 ( $b$ -Flat Local Minima). Given a real-valued objective function $\mathcal { L } ( z ; \theta )$ , for any $b > 0$ , $\theta ^ { \star }$ is a $b$ -flat local minima of $\mathcal { L } ( z ; \theta )$ , if the following conditions are satisfied.
+
+• Condition 1: $\mathcal { L } ( z ; \theta ^ { \star } ) = \mathcal { L } ( z ; \theta ^ { \star } + \epsilon )$ , where $- \mathbf { b } \preceq \epsilon \preceq \mathbf { b }$ and $\mathbf { b } _ { i } = b$ .
+
+• Condition 2: there exist $\mathbf { c } _ { 1 } \prec \theta ^ { \star } - b$ and ${ \bf c } _ { 2 } \succ \theta ^ { \star } + b ,$ , s.t. $\mathcal { L } ( z ; \theta ) > \mathcal { L } ( z ; \theta ^ { \star } )$ , where $\mathbf { c } _ { 1 } \prec \theta \prec \theta ^ { \star } - b$ and $\mathcal { L } ( z ; \theta ^ { \star } ) < \mathcal { L } ( z ; \theta )$ , where $\theta ^ { \star } + b \prec \theta \prec \mathbf { c } _ { 2 }$ .
+
+![](images/4aa02a309c91762f668ef9d87403ca2452fa0984b7055d381e241148d4136b17.jpg)  
+Figure 2: Illustration of our approach and existing solutions. indicates the incremental learning steps on new classes. $R _ { 1 }$ and $R _ { 2 }$ respectively denote the loss of base classes before and after minimizing the loss of new classes. (a) SGD finds sharp minima in the base training. Directly tuning the model on new classes will result in a severe performance drop on base classes. (b) Enforcing strong constraints on parameters by penalizing parameter changes [1, 28, 34] may still lead to a significant performance drop on base classes. (c) Finding flat local minima of base classes and clamping the parameters after trained on new classes to make them fall within the flat region can effectively mitigate catastrophic forgetting.
+
+In practice, it is hard to find the flat local minima that strictly satisfies the above definition, which may not even exist. Hence, our goal is to find an approximately flat local minima of the base training objective function. To this end, we propose to add some small random noise to the model parameters. The noise can be added for multiple times to obtain similar but different loss functions, which will be optimized together to locate the flat minima region. The intuition is clear – the parameter vectors around the flat local minima also have small function values.
+
+To formally state the idea, we assume that the model is parameterized by $\theta = \{ \phi , \psi \}$ , where $\phi$ denotes the parameters of the embedding network and $\psi$ denotes the parameters of the classifier. $z$ denotes a labelled training sample. Denote the loss function by $\mathcal { L } \colon \dot { \mathbb { R } } ^ { d _ { z } }  \mathbb { R }$ . Our target is to minimize the expected loss function $R$ $\ ? \colon \mathbb { R } ^ { d }  \mathbb { R }$ w.r.t. the joint distribution of data $z$ and noise $\epsilon$ , i.e.,
+
+$$
+R ( \theta ) = \int _ { \mathbb { R } ^ { d _ { \epsilon } } } \int _ { \mathbb { R } ^ { d _ { z } } } \mathcal { L } ( z ; \phi + \epsilon , \psi ) d P ( z ) d P ( \epsilon ) = \mathbb { E } [ \mathcal { L } ( z ; \phi + \epsilon , \psi ) ] ,
+$$
+
+where $P ( z )$ is the data distribution and $P ( \epsilon )$ is the noise distribution, and $z$ and $\epsilon$ are independent. Since it is impossible to minimize the expected loss, we minimize its estimation, the empirical loss, which is given by
+
+$$
+\mathcal { L } ( \theta ) = \frac { 1 } { M } \sum _ { j = 1 } ^ { M } \mathcal { L } _ { \mathrm { b a s e } } ( z ; \phi + \epsilon _ { j } , \psi ) , \mathrm { w h e r e }
+$$
+
+$$
+\mathcal { L } _ { \mathrm { b a s e } } ( z ; \phi + \epsilon _ { j } , \psi ) = \frac { 1 } { | \mathcal { D } ^ { 1 } | } \sum _ { z \in \mathcal { D } ^ { 1 } } \mathcal { L } _ { c e } ( z ; \phi + \epsilon _ { j } , \psi ) + \lambda \frac { 1 } { | \mathcal { C } ^ { 1 } | } \sum _ { c \in \mathcal { C } ^ { 1 } } \| p _ { c } - p _ { c } ^ { * } \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\epsilon _ { j }$ is a noise vector sampled from $P ( \epsilon )$ , $M$ is the sampling times, $\mathcal { L } _ { c e } ( z ; \phi + \epsilon _ { j } , \psi )$ refers to the cross-entropy loss of a training sample $z$ , and $p _ { c }$ and $p _ { c } ^ { * }$ are the class prototypes before and after injecting noise respectively. The first term of $\mathcal { L } _ { b a s e }$ is designed to find the flat region where the parameters $\phi$ of the embedding network can well separate the base classes. The second term enforces the class prototypes fixed within such region, which is designed to solve the prototype drift problem [54, 8] (the class prototypes change after updating the network) in later incremental learning sessions such that the saved base class prototypes can be directly used for evaluation in later sessions.
+
+# 4.2 Incremental Few-shot Learning within the Flat Region
+
+In the incremental few-shot learning sessions $\left( t \geq 2 \right)$ ), we fine-tune the parameters $\phi$ of the embedding network within the flat region to learn new classes. It is worth noting that while the flat region might be relatively small, it is enough for incremental few-shot learning. Because only few training samples are provided for each new class, to prevent overfitting in few-shot learning, excessive training should be avoided and only a small number of update iterations should be applied.
+
+# Algorithm 1: F2M
+
+Input: the flat region bound $b$ , randomly initialized $\theta = \{ \phi , \psi \}$ , the step sizes $\alpha$ and $\beta$ .   
+// Training over base classes $t = 1$   
+for epoch $k = 1 , 2 , \dots$ do for $j = 1 , 2 , . . . , M$ do Sample a noise vector $\epsilon _ { j } \sim P ( \epsilon )$ , s.t. $- \mathbf { b } \preceq \epsilon _ { j } \preceq \mathbf { b }$ ; Add the noise to the parameters of the embedding network, i.e., $\theta = \{ \phi + \epsilon _ { j } , \psi \}$ ; Compute the base loss $\mathcal { L } _ { b a s e }$ with Eq. 4; Reset the parameters, i.e., $\theta = \{ \phi , \psi \}$ ; end Update $\theta = \theta - \alpha \nabla { \mathcal { L } } ( \theta )$ with the loss $\mathcal { L }$ defined in Eq. 3.   
+end   
+Normalize and save the prototype of each base class;   
+// Incremental learning $t \geq 2$   
+Combine the training data $\mathcal { D } ^ { t }$ and the exemplars saved in previous few-shot sessions $2 \leq t _ { e } < t$ ;   
+for epoch $k = 1 , 2 , . .$ . do Compute the metric-based classification loss ${ \mathcal { L } } _ { m }$ by Eq. 5; Update $\phi = \phi - \beta \nabla \mathcal { L } _ { m } ( z ; \phi )$ ; Clamp the parameters $\phi$ to ensure they fall in the flat minima region;   
+end   
+Randomly select and save a few exemplars from the training data $\mathcal { D } ^ { t }$ ;   
+Normalize and save the prototype of each new class;   
+Output: Model parameters $\theta = \{ \phi , \psi \}$ .
+
+We employ a metric-based classification algorithm with Euclidean distance to fine-tune the parameters. The loss function is defined as
+
+$$
+\mathcal { L } _ { m } ( z ; \phi ) = - \sum _ { z \in \mathcal { D } } \sum _ { c \in \mathcal { C } } \mathbb { 1 } ( y = c ) \log \big ( \frac { e ^ { - d ( p _ { c } , f ( x ; \phi ) ) } } { \sum _ { c _ { k } \in \mathcal { C } } e ^ { - d ( p _ { c _ { k } } , f ( x ; \phi ) ) } } \big ) , z
+$$
+
+where $d ( \cdot , \cdot )$ denotes Euclidean distance, $p _ { c }$ is the prototype of class $c$ , $\textstyle { \mathcal { C } } = \bigcup _ { i = 1 } ^ { t } { \mathcal { C } } ^ { i }$ refers to all encountered classes, and ${ \mathcal { D } } = { \mathcal { D } } ^ { t } \cup { \mathcal { P } }$ denotes the union of the current training data $\mathcal { D } ^ { t }$ and the exemplar set $\mathcal { P } = \{ P _ { 2 } , . . . , P _ { t - 1 } \}$ , where $P _ { t _ { e } } ( 2 \leq t _ { e } < t )$ is the set of saved exemplars in session $t _ { e }$ . Note that the prototypes of new classes are computed by Eq. 1, and those of base classes are saved in the base session. After updating the embedding network parameters, we clamp them to ensure that they fall within the flat region, i.e. $\phi ^ { \star } - b \preceq \phi \preceq \phi ^ { \star } + b$ , where $\phi ^ { \star }$ denotes the optimal parameter vector learned in the base session. After fine-tuning, we evaluate the model using the nearest class mean classifier as in Eq. 1, with previously saved prototypes and newly computed ones. The whole training process is described in Algorithm 1. Note that to calibrate the estimates of the classifier, we normalize all prototypes to make those of base classes and those of new classes have the same norm.
+
+# 4.3 Convergence Analysis
+
+Our aim is to find a flat region within which all parameter vectors work well. We then minimize the expected loss w.r.t. the joint distribution of noise $\epsilon$ and data $z$ . To approximate this expected loss, we sample from $P ( \epsilon )$ for multiple times in each iteration and optimize the objective function using stochastic gradient descent (SGD). Here, we provide theoretical guarantees for our method. Given the non-convex loss function in Eq. 3, we prove the convergence of our proposed method. The proof idea is inspired by the convergence analysis of SGD [3, 27].
+
+Formally, in each batch $k$ , let $z _ { k }$ denote the batch data, $\{ \epsilon _ { j } \} _ { j = 1 } ^ { M }$ be the sampled noises, and $\alpha _ { k }$ be the step size. In the base training session, we update the model parameters as follows:
+
+$$
+\theta _ { k + 1 } = \theta _ { k } - \frac { \alpha _ { k } } { M } \sum _ { j = 1 } ^ { M } \nabla \mathcal { L } _ { \mathrm { b a s e } } \big ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } \big ) = \theta _ { k } - \frac { \alpha _ { k } } { M } \sum _ { j = 1 } ^ { M } g \big ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } \big ) ,
+$$
+
+where $g ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } ) = \nabla { \mathcal { L } } _ { \mathrm { b a s e } } ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } )$ is the gradient. To formally analyze the convergence of our algorithm, we define the following assumptions.
+
+Assumption 4.1 (L-smooth risk function). The expected loss function $R : \mathbb { R } ^ { d }  \mathbb { R }$ (Eq. 2) is continuously differentiable and $L$ -smooth with constant $L > 0$ such that
+
+$$
+\| \nabla R ( \theta ) - \nabla R ( \theta ^ { \prime } ) \| _ { 2 } \leq L \| \theta - \theta ^ { \prime } \| .
+$$
+
+This assumption is significant for the convergence analysis of gradient-based optimization algorithms, since it limits how fast the gradient of the loss function can change w.r.t. the parameter vector.
+
+Assumption 4.2. The expected loss function satisfies the following conditions:
+
+• Condition 1: $R$ is bounded below by a scalar $R ^ { \star }$ , given the sequence of parameters $\{ \theta _ { k } \}$
+
+• Condition 2: For all $k \in \mathbb N$ and $j \in [ 1 , M ]$ ,
+
+$$
+\begin{array} { r } { \mathbb { E } _ { z _ { k } , \epsilon _ { j } } \big [ g \big ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } \big ) \big ] = \nabla R ( \theta _ { k } ) . } \end{array}
+$$
+
+• Condition $^ 3$ : There exist scalars $m _ { 1 } \geq 0$ and $m _ { 2 } \geq 0 ,$ , for all $k \in \mathbb N$ and $j \in [ 1 , M ]$ ,
+
+$$
+\begin{array} { r } { \mathbb { V } _ { z _ { k } , \epsilon _ { j } } [ g ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } ) ] \le m _ { 1 } + m _ { 2 } \| \nabla R ( \theta _ { k } ) \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+$\mathbb { E } _ { z _ { k } , \epsilon _ { j } } [ \cdot ]$ denotes the expectation w.r.t. the joint distribution of random variables $z _ { k }$ and $\epsilon _ { j }$ , and $\mathbb { V } _ { z _ { k } , \epsilon _ { j } } [ \cdot ]$ denotes the variance. Condition 1 ensures that the expected loss $R$ is bounded by a minimum value $R ^ { \star }$ during the updates, which is a natural and practical assumption. Condition 2 assumes that the gradient $g ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } )$ is an unbiased estimate of $\nabla R ( \theta _ { k } )$ . This is a strict assumption made to simplify the proof, but it can be easily relaxed to a general and easily-met condition that there exist $\mu _ { 1 } \geq \mu _ { 2 } > 0$ satisfying $\| \mathbb { E } _ { z _ { k } , \epsilon _ { j } } [ \bar { g ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } ) } ] \| _ { 2 } \leq \mu _ { 1 } \| \nabla \bar { R ( \theta _ { k } ) } \| _ { 2 }$ and $\nabla R ( \theta _ { k } ) ^ { T } \mathbb { E } _ { z _ { k } , \epsilon _ { j } } [ g ( \underline { { z _ { k } } } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } ) ] \geq \mu _ { 2 } \| \nabla R ( \theta _ { k } ) \| _ { 2 } ^ { 2 }$ . Therefore, the convergence can be proved in a similar way using the techniques presented in the Appendix. Condition 3 assumes the variance of the gradient $g ( z _ { k } ; \phi _ { k } + \epsilon _ { j } , \psi _ { k } )$ cannot be arbitrarily large, which is also reasonable in practice. To facilitate later analysis, similar to [43], we restrict the step sizes as follows.
+
+Assumption 4.3. The learning rates satisfy:
+
+$$
+\sum _ { k = 1 } ^ { \infty } \alpha _ { k } = \infty , \ \sum _ { k = 1 } ^ { \infty } \alpha _ { k } ^ { 2 } < \infty .
+$$
+
+This assumption can be easily met, since in practice the learning rate $\alpha _ { k }$ is usually far less than 1 and decreases w.r.t. $k$ . Based on the above assumptions, we can derive the following theorem.
+
+Theorem 4.1. Under assumptions 4.1, 4.2 and 4.3, we further assume that the risk function $R$ is twice differentiable, and that $\| \nabla R ( \theta ) \| _ { 2 } ^ { 2 }$ is $L _ { 2 }$ -smooth with constant $L _ { 2 } > 0$ , then we have
+
+$$
+\operatorname* { l i m } _ { k \to \infty } \mathbb { E } [ \| \nabla R ( \theta _ { k } ) \| _ { 2 } ^ { 2 } ] = 0 .
+$$
+
+This theorem establishes the convergence of our algorithm. The proof is provided in Appendix A.1.
+
+# 5 Experiments
+
+In this section, we empirically evaluate our proposed method for incremental few-shot learning and demonstrate its effectiveness by comparison with state-of-the-art methods.
+
+# 5.1 Experimental Setup
+
+Datasets. For CIFAR-100 and miniImageNet, we randomly select 60 classes as the base classes and the remaining 40 classes as the new classes. In each incremental learning session, we construct 5-way 5-shot tasks by randomly picking 5 classes and sampling 5 examples for each class. For CUB-200-2011 with 200 classes, we select 100 classes as the base classes and 100 classes as the new ones. We test 10-way 5-shot tasks on this dataset.
+
+Baselines. We compare our method F2M with 8 methods: the Baseline proposed in Sec. 3, a jointtraining method that uses all previously seen data including the base and the following few-shot tasks for training, the classifier re-training method (cRT) [25] for long-tailed classification trained with all encountered data, iCaRL [40], Rebalance [20], TOPIC [47], FSLL [34], and IDLVQ-C [8]. For a fair comparison, we re-implement cRT [25], iCaRL [40], Rebalance [20], FSLL [34], and the joint-training method and tune them to their best performance. We also provide the results reported in the original papers for comparison. The results of TOPIC [47] and IDLVQ-C [8] are copied from the original papers. Note that for IL, joint-training is naturally the upper bound of incremental learning algorithms, however, for IFL, joint-training is not a good approximation of the upper bound because data imbalance makes the model perform significantly poorer on new classes (long-tailed classes). To address the data imbalance issue, we re-implement the cRT method as the approximate upper bound.
+
+Table 1: Classification accuracy on CIFAR-100 for 5-way 5-shot incremental learning. ∗ indicates our re-implementation.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="9">sessions</td><td rowspan="2">The gap with cRT</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>cRT[25]*</td><td>65.18</td><td>63.89</td><td>60.20</td><td>57.23</td><td>53.71</td><td>50.39</td><td>48.77</td><td>47.29</td><td>45.28</td><td>=</td></tr><tr><td>Joint-training*</td><td>65.18</td><td>61.45</td><td>57.36</td><td>53.68</td><td>50.84</td><td>47.33</td><td>44.79</td><td>42.62</td><td>40.08</td><td>-5.20</td></tr><tr><td>Baseline</td><td>65.18</td><td>61.67</td><td>58.61</td><td>55.11</td><td>51.86</td><td>49.43</td><td>47.60</td><td>45.64</td><td>43.83</td><td>-1.45</td></tr><tr><td>iCaRL[40]*</td><td>66.52</td><td>57.26</td><td>54.27</td><td>50.62</td><td>47.33</td><td>44.99</td><td>43.14</td><td>41.16</td><td>39.49</td><td>-5.79</td></tr><tr><td>Rebalance [20]*</td><td>66.66</td><td>61.42</td><td>57.29</td><td>53.02</td><td>48.85</td><td>45.68</td><td>43.06</td><td>40.56</td><td>38.35</td><td>-6.93</td></tr><tr><td>FSLL [34]*</td><td>65.18</td><td>56.24</td><td>54.55</td><td>51.61</td><td>49.11</td><td>47.27</td><td>45.35</td><td>43.95</td><td>42.22</td><td>-3.08</td></tr><tr><td>iCaRL[40]</td><td>64.10</td><td>53.28</td><td>41.69</td><td>34.13</td><td>27.93</td><td>25.06</td><td>20.41</td><td>15.48</td><td>13.73</td><td>-31.55</td></tr><tr><td>Rebalance [20]</td><td>64.10</td><td>53.05</td><td>43.96</td><td>36.97</td><td>31.61</td><td>26.73</td><td>21.23</td><td>16.78</td><td>13.54</td><td>-31.74</td></tr><tr><td>TOPIC [47]</td><td>64.10</td><td>55.88</td><td>47.07</td><td>45.16</td><td>40.11</td><td>36.38</td><td>33.96</td><td>31.55</td><td>29.37</td><td>-15.91</td></tr><tr><td>FSLL [34]</td><td>64.10</td><td>55.85</td><td>51.71</td><td>48.59</td><td>45.34</td><td>43.25</td><td>41.52</td><td>39.81</td><td>38.16</td><td>-7.12</td></tr><tr><td>FSLL+SS [34]</td><td>66.76</td><td>55.52</td><td>52.20</td><td>49.17</td><td>46.23</td><td>44.64</td><td>43.07</td><td>41.20</td><td>39.57</td><td>-5.71</td></tr><tr><td>F2M</td><td>64.71</td><td>62.05</td><td>59.01</td><td>55.58</td><td>52.55</td><td>49.96</td><td>48.08</td><td>46.28</td><td>44.67</td><td>-0.61</td></tr></table>
+
+Table 2: Classification accuracy on miniImageNet for 5-way 5-shot incremental learning. ∗ indicates our re-implementation.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="9">sessions</td><td rowspan="2">The gap with cRT</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>cRT[25]*</td><td>67.30</td><td>64.15</td><td>60.59</td><td>57.32</td><td>54.22</td><td>51.43</td><td>48.92</td><td>46.78</td><td>44.85</td><td>=</td></tr><tr><td>Joint-training*</td><td>67.30</td><td>62.34</td><td>57.79</td><td>54.08</td><td>50.93</td><td>47.65</td><td>44.64</td><td>42.61</td><td>40.29</td><td>-4.56</td></tr><tr><td>Baseline</td><td>67.30</td><td>63.18</td><td>59.62</td><td>56.33</td><td>53.28</td><td>50.50</td><td>47.96</td><td>45.85</td><td>43.88</td><td>-0.97</td></tr><tr><td>iCaRL [40]*</td><td>67.35</td><td>59.91</td><td>55.64</td><td>52.60</td><td>49.43</td><td>46.73</td><td>44.13</td><td>42.17</td><td>40.29</td><td>-4.56</td></tr><tr><td>Rebalance [20]*</td><td>67.91</td><td>63.11</td><td>58.75</td><td>54.83</td><td>50.68</td><td>47.11</td><td>43.88</td><td>41.19</td><td>38.72</td><td>-6.13</td></tr><tr><td>FSLL [34]*</td><td>67.30</td><td>59.81</td><td>57.26</td><td>54.57</td><td>52.05</td><td>49.42</td><td>46.95</td><td>44.94</td><td>42.87</td><td>-1.11</td></tr><tr><td>iCaRL[40]</td><td>61.31</td><td>46.32</td><td>42.94</td><td>37.63</td><td>30.49</td><td>24.00</td><td>20.89</td><td>18.80</td><td>17.21</td><td>-27.64</td></tr><tr><td>Rebalance [20]</td><td>61.31</td><td>47.80</td><td>39.31</td><td>31.91</td><td>25.68</td><td>21.35</td><td>18.67</td><td>17.24</td><td>14.17</td><td>-30.68</td></tr><tr><td>TOPIC [47]</td><td>61.31</td><td>50.09</td><td>45.17</td><td>41.16</td><td>37.48</td><td>35.52</td><td>32.19</td><td>29.46</td><td>24.42</td><td>-20.43</td></tr><tr><td>FSLL [34]</td><td>66.48</td><td>61.75</td><td>58.16</td><td>54.16</td><td>51.10</td><td>48.53</td><td>46.54</td><td>44.20</td><td>42.28</td><td>-2.57</td></tr><tr><td>FSLL+SS[34]</td><td>68.85</td><td>63.14</td><td>59.24</td><td>55.23</td><td>52.24</td><td>49.65</td><td>47.74</td><td>45.23</td><td>43.92</td><td>-0.93</td></tr><tr><td>IDLVQ-C[8]</td><td>64.77</td><td>59.87</td><td>55.93</td><td>52.62</td><td>49.88</td><td>47.55</td><td>44.83</td><td>43.14</td><td>41.84</td><td>-3.01</td></tr><tr><td>F2M</td><td>67.28</td><td>63.80</td><td>60.38</td><td>57.06</td><td>54.08</td><td>51.39</td><td>48.82</td><td>46.58</td><td>44.65</td><td>-0.20</td></tr></table>
+
+Experimental details. The experiments are conducted with NVIDIA GPU RTX3090 on CUDA 11.0. We randomly split each dataset into multiple tasks (sessions). For each dataset (with a fixed split), we run each algorithm for 10 times and report the mean accuracy. We adopt ResNet18 [16] as the backbone network. For data augmentation, we use standard random crop and horizontal flip. In the base training stage, we select the last 4 or 8 convolution layers to inject noise, because these layers output higher-level feature representations. The flat region bound $b$ is set as 0.01. We set the number of times for noise sampling as $M = 2 \sim 4$ , since a larger $M$ will increase the training time. In each incremental few-shot learning session, the total number of training epochs is 6, and the learning rate is 0.02. To verify the correctness of our implementation, we conduct experiments on incremental learning and compare our results to those reported on CIFAR-100 in Appendix A.3. More experiment details are provided in Appendix A.2.
+
+Table 3: Classification accuracy on CUB-200-2011 for 10-way 5-shot incremental learning.∗ indicates our re-implementation.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">sessions</td><td rowspan="2">The gap with cRT</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>cRT[25]*</td><td>80.83</td><td>78.51</td><td>76.12</td><td>73.93</td><td>71.46</td><td>68.96</td><td>67.73</td><td>66.75</td><td>64.22</td><td>62.53</td><td>61.08</td><td></td></tr><tr><td>Joint-training*</td><td>80.83</td><td>77.57</td><td>74.11</td><td>70.75</td><td>68.52</td><td>65.97</td><td>64.58</td><td>62.22</td><td>60.18</td><td>58.49</td><td>56.78</td><td>-4.30</td></tr><tr><td>Baseline</td><td>80.87</td><td>77.15</td><td>74.46</td><td>72.26</td><td>69.47</td><td>67.18</td><td>65.62</td><td>63.68</td><td>61.30</td><td>59.72</td><td>58.12</td><td>-2.96</td></tr><tr><td>iCaRL [40]*</td><td>79.58</td><td>67.63</td><td>64.17</td><td>61.80</td><td>58.10</td><td>55.51</td><td>53.34</td><td>50.89</td><td>48.62</td><td>47.34</td><td>45.60</td><td>-15.48</td></tr><tr><td>Rebalance [20]*</td><td>80.94</td><td>70.32</td><td>62.96</td><td>57.19</td><td>51.06</td><td>46.70</td><td>44.03</td><td>40.15</td><td>36.75</td><td>34.88</td><td>32.09</td><td>-28.99</td></tr><tr><td>FSLL [34]*</td><td>80.83</td><td>77.38</td><td>72.37</td><td>71.84</td><td>67.51</td><td>65.30</td><td>63.75</td><td>61.16</td><td>59.05</td><td>58.03</td><td>55.82</td><td>-5.26</td></tr><tr><td>iCaRL [40]</td><td>68.68</td><td>52.65</td><td>48.61</td><td>44.16</td><td>36.62</td><td>29.52</td><td>27.83</td><td>26.26</td><td>24.01</td><td>23.89</td><td>21.16</td><td>-39.92</td></tr><tr><td>Rebalance [20]</td><td>68.68</td><td>57.12</td><td>44.21</td><td>28.78</td><td>26.71</td><td>25.66</td><td>24.62</td><td>21.52</td><td>20.12</td><td>20.06</td><td>19.87</td><td>-41.21</td></tr><tr><td>TOPIC[47]</td><td>68.68</td><td>62.49</td><td>54.81</td><td>49.99</td><td>45.25</td><td>41.40</td><td>38.35</td><td>35.36</td><td>32.22</td><td>28.31</td><td>26.28</td><td>-34.80</td></tr><tr><td>FSLL [34]</td><td>72.77</td><td>69.33</td><td>65.51</td><td>62.66</td><td>61.10</td><td>58.65</td><td>57.78</td><td>57.26</td><td>55.59</td><td>55.39</td><td>54.21</td><td>-6.87</td></tr><tr><td>FSLL+SS [34]</td><td>75.63</td><td>71.81</td><td>68.16</td><td>64.32</td><td>62.61</td><td>60.10</td><td>58.82</td><td>58.70</td><td>56.45</td><td>56.41</td><td>55.82</td><td>-5.26</td></tr><tr><td>IDLVQ-C [8]</td><td>77.37</td><td>74.72</td><td>70.28</td><td>67.13</td><td>65.34</td><td>63.52</td><td>62.10</td><td>61.54</td><td>59.04</td><td>58.68</td><td>57.81</td><td>-3.27</td></tr><tr><td>F2M</td><td>81.07</td><td>78.16</td><td>75.57</td><td>72.89</td><td>70.86</td><td>68.17</td><td>67.01</td><td>65.26</td><td>63.36</td><td>61.76</td><td>60.26</td><td>-0.82</td></tr></table>
+
+Table 4: Comparison of the flatness of the local minima found by the Baseline and our F2M.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Indicator I</td><td colspan="2">Variance g2</td></tr><tr><td>Training Set</td><td>Testing Set</td><td>Training Set</td><td>Testing Set</td></tr><tr><td>Baseline</td><td>0.2993</td><td>0.4582</td><td>0.1451</td><td>0.2395</td></tr><tr><td>F2M</td><td>0.0506</td><td>0.0800</td><td>0.0296</td><td>0.0334</td></tr></table>
+
+# 5.2 Comparison with the State-of-the-Art
+
+F2M outperforms the state-of-the-art methods. The main results on CIFAR-100, miniImageNet and CUB-200-2011 are presented in Table 1, Table 2 and Table 3 respectively. Based on the experiment results, we have the following observations: 1) The Baseline introduced in Sec. 3 outperforms the state-of-the-art approaches on all incremental sessions. 2) As expected, cRT consistently outperforms the Baseline up to $1 \%$ to $3 \%$ by considering the data imbalance problem and applying proper techniques to tackle the long-tailed classification problem to improve performance. Hence, it is reasonable to use cRT as the approximate upper bound of IFL. 3) Our F2M outperforms the state-ofthe-art methods and the Baseline. Moreover, the performance of F2M is very close to the approximate upper bound, i.e., the gap with cRT is only $0 . 2 \%$ in the last session on miniImageNet. The results show that even with strong constraints [20, 40, 34] and saved examplars of base classes [20, 40, 8], current methods cannot effectively address the catastrophic forgetting problem. In contrast, finding flat minima seems a promising approach to overcome this harsh problem.
+
+# 5.3 Ablation Study and Analysis
+
+Analysis on the flatness of local minima. Here, we verify that our method can find a more flat local minima than the Baseline. For a found local minima $\theta ^ { \star }$ , we measure its flatness as follows. We sample the noise for 1000 times. For each time, we inject the sampled noise to $\theta ^ { \star }$ and calculate the loss $\mathcal { L } _ { i }$ . to measure the flatness. Then, we adopt the indicator I = 11000 P1000i=1 $\mathcal { L } ^ { * }$ denotes the loss of $\begin{array} { r } { I = \frac { 1 } { 1 0 0 0 } \sum _ { i = 1 } ^ { 1 0 0 0 } ( \mathcal { L } _ { i } - \mathcal { L } ^ { * } ) ^ { 2 } } \end{array}$ $\theta ^ { \star }$ , and $\overline { { \mathcal { L } } }$ 1000denotes the average loss of and variance $\begin{array} { r } { \sigma ^ { 2 } = \frac { 1 } { 1 0 0 0 } \sum _ { i = 1 } ^ { 1 0 0 0 } ( \mathcal { L } _ { i } - \overline { { \mathcal { L } } } ) ^ { 2 } } \end{array}$ $\{ \mathcal { L } _ { i } \} _ { i = 1 } ^ { 1 0 0 0 }$ . The values of the indicator and variance of F2M and the Baseline are presented in Table 4, which clearly demonstrate that our method can find a more flat local minima.
+
+Ablation study on the designs of our method. Here, we study the effectiveness of each design of our method, including adding noise to the model parameters for finding $b$ -flat local minima (FM) during the base training session, the prototype fixing term (PF) used in the base training objective (Eq. 4), parameter clamping (PC) during incremental learning, and prototype normalization (PN). We conduct an ablation study by removing each component in turn and report the experimental results in Table 5.
+
+Finding b-flat local minima. Standard supervised training with SGD as the optimizer tends to converge to a sharp local minima. It leads to a significant drop in performance because the loss changes quickly in the neighborhood of the sharp local minima. As shown in Table 5, even with parameter clamping during incremental learning, the performance still drops significantly. In contrast, restricting the parameters in a small flat region can mitigate the forgetting problem.
+
+Table 5: Ablation study of our F2M on CIFAR-100. PD refers to the performance dropping rate.   
+
+<table><tr><td rowspan="2">FM</td><td rowspan="2">PF</td><td rowspan="2">PC</td><td rowspan="2">PN</td><td colspan="10">sessions</td><td rowspan="2">PD↓</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td rowspan="4">广</td><td></td><td></td><td></td><td>65.18</td><td>60.83</td><td>53.13</td><td>43.57</td><td>23.75</td><td>10.76</td><td>08.26</td><td>07.24</td><td>06.45</td><td>58.73</td></tr><tr><td></td><td>√</td><td></td><td>65.18</td><td>59.48</td><td>56.77</td><td>52.99</td><td>50.09</td><td>47.80</td><td>45.92</td><td>44.20</td><td>42.55</td><td>22.63</td></tr><tr><td>√</td><td></td><td>√</td><td>64.71</td><td>59.54</td><td>53.03</td><td>45.09</td><td>41.68</td><td>39.04</td><td>38.64</td><td>37.19</td><td>36.01</td><td>28.70</td></tr><tr><td></td><td>√</td><td>√</td><td>64.55</td><td>61.27</td><td>58.33</td><td>54.82</td><td>51.60</td><td>49.22</td><td>47.48</td><td>45.78</td><td>44.08</td><td>20.47</td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td>64.71</td><td>61.75</td><td>58.80</td><td>55.33</td><td>52.27</td><td>49.75</td><td>47.72</td><td>46.01</td><td>44.43</td><td>20.28</td></tr><tr><td>√</td><td></td><td>√</td><td>√</td><td>64.71</td><td>61.99</td><td>58.99</td><td>55.58</td><td>52.55</td><td>49.96</td><td>48.08</td><td>46.28</td><td>44.67</td><td>20.04</td></tr></table>
+
+Table 6: Study of the flat region bound $b$ for 5-way 5-shot incremental learning on CIFAR-100. The top 3 results in each row are in boldface.   
+
+<table><tr><td rowspan="2">Session</td><td colspan="6">The hyperparameter b</td></tr><tr><td>0.0025</td><td>0.005</td><td>0.01</td><td>0.02</td><td>0.04</td><td>0.08</td></tr><tr><td>Session 1(60 bases classes)</td><td>64.85</td><td>64.67</td><td>64.81</td><td>64.71</td><td>63.30</td><td>62.25</td></tr><tr><td>Session 9 (All 100 classes)</td><td>44.16</td><td>44.54</td><td>44.58</td><td>44.67</td><td>43.75</td><td>43.04</td></tr><tr><td>Session 9 (60 base classes)</td><td>59.58</td><td>59.69</td><td>59.73</td><td>59.44</td><td>58.38</td><td>57.21</td></tr><tr><td>Session 9 (40 new classes)</td><td>21.03</td><td>21.81</td><td>21.86</td><td>22.52</td><td>21.80</td><td>21.77</td></tr></table>
+
+Prototype fixing. Without fixing the prototypes after injecting noise to selected layers during the process of finding local minima, i.e. removing the second term of Eq. 4, it is still possible to tune the model within the flat region to well separate base classes. However, the saved prototypes of base classes will become less accurate because the embeddings of the base samples suffer from semantic drift [54]. As shown in Table 5, it results in a performance drop of nearly $0 . 6 \%$ .
+
+Parameter clamping. Parameter clamping restricts the model parameters to the $b$ -flat region after incremental few-shot learning. Outside the $b$ -flat region, the performance drops quickly. It can be seen from Table 5 that removing parameter clamping leads to a significant drop in performance.
+
+Prototype normalization. As mentioned in Sec. 4.2, we normalize the class prototypes to calibrate the estimates of the class mean classifier. The results in Table 5 show the effectiveness of normalization, which helps to further improve the performance.
+
+Study of the flat region bound $b$ . We study the effect of the flat region bound $b$ for 5-way 5-shot incremental learning on CIFAR-100. We report the test accuracy in session 1 (base session) and session 9 (last session) w.r.t. different $b$ in Table 6. It can be seen that the best results are achieved for $b \in [ 0 . 0 0 5 , 0 . 0 2 ]$ . A larger $b$ (e.g., 0.04 or 0.08) leads to a significant performance drop on base classes, even for those in session 1, indicating that there may not exist a large flat region around a good local minima. Meanwhile, a smaller $b$ (e.g., 0.0025) results in a performance decline on new classes, due to the overly small capacity of the flat region. This illustrates the trade-off effect of $b$ .
+
+# 6 Conclusion
+
+We have proposed a novel approach to overcome catastrophic forgetting in incremental few-shot learning by finding flat local minima of the objective function in the base training stage and then fine-tuning the model within the flat region on new tasks. Extensive experiments on benchmark datasets show that our model can effectively mitigate catastrophic forgetting and adapt to new classes. A limitation of our method is that it may not be suitable for medium- or high-shot tasks, since the flat region is relatively small, which limits the model capacity. However, it is still possible to adapt our core idea for incremental learning. For example, one can search for a less flat but wider local minima region in the base training stage and tune the model within this region during incremental learning sessions, where previous techniques such as elastic weight consolidation (EWC) [28] can be used to constraint the model parameters. This could be an interesting direction for future research.
+
+# Acknowledgments and Disclosure of Funding
+
+We would like to thank the anonymous reviewers for their insightful and helpful comments. This research was supported by the grant of DaSAIL project P0030935 funded by PolyU/UGC.
+
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+[58] Chi Zhang, Yujun Cai, Guosheng Lin, and Chunhua Shen. Deepemd: Few-shot image classification with differentiable earth mover’s distance and structured classifiers. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12203–12213, 2020.   
+[59] Chi Zhang, Nan Song, Guosheng Lin, Yun Zheng, Pan Pan, and Yinghui Xu. Few-shot incremental learning with continually evolved classifiers. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12455–12464, 2021.   
+[60] Hanbin Zhao, Yongjian Fu, Mintong Kang, Q Tian, F Wu, and X Li. Mgsvf: Multi-grained slow vs. fast framework for few-shot class-incremental learning. arXiv preprint arXiv:2006.15524.
+
+# 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 Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 4.3.   
+(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The link to the source code is provided in the abstract.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 5.1 and Appendix A.2.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Appendix A.3.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 5.1
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We have cited [29, 49, 50].   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/ALvt7nXa2q/ALvt7nXa2q_content_list.json

@@ -0,0 +1,1568 @@
+[
+    {
+        "type": "text",
+        "text": "Overcoming Catastrophic Forgetting in Incremental Few-Shot Learning by Finding Flat Minima ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Guangyuan Shi,∗ Jiaxin Chen∗, Wenlong Zhang, Li-Ming Zhan, Xiao-Ming Wu† Department of Computing The Hong Kong Polytechnic University {guang-yuan.shi, jiax.chen, wenlong.zhang, lmzhan.zhan}@connect.polyu.hk xiao-ming.wu@polyu.edu.hk ",
+        "bbox": [
+            187,
+            224,
+            808,
+            296
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            330,
+            535,
+            348
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper considers incremental few-shot learning, which requires a model to continually recognize new categories with only a few examples provided. Our study shows that existing methods severely suffer from catastrophic forgetting, a well-known problem in incremental learning, which is aggravated due to data scarcity and imbalance in the few-shot setting. Our analysis further suggests that to prevent catastrophic forgetting, actions need to be taken in the primitive stage – the training of base classes instead of later few-shot learning sessions. Therefore, we propose to search for flat local minima of the base training objective function and then fine-tune the model parameters within the flat region on new tasks. In this way, the model can efficiently learn new classes while preserving the old ones. Comprehensive experimental results demonstrate that our approach outperforms all prior state-of-the-art methods and is very close to the approximate upper bound. The source code is available at https://github.com/moukamisama/F2M. ",
+        "bbox": [
+            233,
+            363,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            568,
+            312,
+            585
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Why study incremental few-shot learning? Incremental learning enables a model to continually learn new concepts from new data without forgetting previously learned knowledge. Rooted from real-world applications, this topic has attracted a significant amount of interest in recent years [5, 30, 31, 40, 26]. Incremental learning assumes sufficient training data is provided for new classes, which is impractical in many application scenarios, especially when the new classes are rare categories which are costly or difficult to collect. This motivates the study of incremental few-shot learning, a more difficult paradigm that aims to continually learn new tasks with only a few examples. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Challenges. The major challenge for incremental learning is catastrophic forgetting [14, 28, 35], which refers to the drastic performance drop on previous tasks after learning new tasks. This phenomenon is caused by the inaccessibility to previous data while learning on new data. Catastrophic forgetting presents a bigger challenge for incremental few-shot learning. Due to the small amount of training data in new tasks, the model tends to severely overfit on new classes while quickly forgetting old classes, resulting in catastrophic performance. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Current research. The study of incremental few-shot learning has just started [47, 41, 60, 9, 8, 34, 59]. Current works mainly borrow ideas from research in incremental learning to overcome the forgetting problem, by enforcing strong constraints on model parameters to penalize the changes of parameters [34, 28, 56], or by saving a small amount of exemplars from old classes and adding constraints on the exemplars to avoid forgetting [40, 20, 4]. However, in our empirical study, we find that an intransigent model that only trains on base classes and does not tune on new tasks consistently outperforms state-of-the-art methods, including a joint-training method [47] that uses all encountered data for training and hence suffers from severe data imbalance. This observation motivates us to address this harsh problem from a different angle. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our solution. Unlike existing solutions that try to overcome the catastrophic forgetting problem during the process of learning new tasks, we adopt a different approach by considering this issue during the training of base classes. Specifically, we propose to search for flat local minima of the base training objective function. For any parameter vector in the flat region around the minima, the loss is small, and the base classes are supposed to be well separated. The flat local minima can be found by adding random noise to the model parameters for multiple times and jointly optimizing multiple loss functions. During the following incremental few-shot learning stage, we fine-tune the model parameters within the flat region, which can be achieved by clamping the parameters after updating them on few-shot tasks. In this way, the model can efficiently learn new classes while preserving the old ones. Our key contributions are summarized as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We conduct a comprehensive empirical study on existing incremental few-shot learning methods and discover that a simple baseline model that only trains on base classes outperforms state-of-the-art methods, which demonstrates the severity of catastrophic forgetting. • We propose a novel approach for incremental few-shot learning by addressing the catastrophic forgetting problem in the primitive stage. Through finding the flat minima region during training on base classes and fine-tuning within the region while learning on new tasks, our model can overcome catastrophic forgetting and avoid overfitting. • Comprehensive experimental results on CIFAR-100, miniImageNet, and CUB-200-2011 show that our approach outperforms all state-of-the-art methods and achieves performance that is very close to the approximate upper bound. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            473,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Few-shot learning aims to learn to generalize to new categories with a few labeled samples in each class. Current few-shot methods mainly include optimization-based methods [12, 23, 32, 39, 45, 46, 55] and metric-based methods [13, 19, 44, 49, 52, 58, 57, 53]. Optimization-based methods can achieve fast adaptation to new tasks with limited samples by learning a specific optimization algorithm. Metric-based approaches exploit different distance metrics such as L2 distance [44], cosine similarity [49], and DeepEMD [58] in the learned metric/embedding space to measure the similarity between samples. Recently, Tian et al. [48] find that standard supervised training can learn a good metric space for unseen classes, which echoes with our observation on the proposed baseline model in Sec. 3. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Incremental learning focuses on the challenging problem of continually learning to recognize new classes in new coming data without forgetting old classes [6, 7, 10, 51]. Previous research mainly includes multi-class incremental learning [4, 38, 22, 33, 54, 51] and multi-task incremental learning [21, 31, 42]. To overcome the catastrophic forgetting problem, some attempts propose to impose strong constraints on model parameters by penalizing the changes of parameters [28, 1]. Other attempts try to enforce constraints on the exemplars of old classes by restricting the output logits [40] or penalizing the changes of embedding angles [20]. In this work, our empirical study shows that imposing strong constraints on the arriving new classes may not be a promising way to tackle incremental few-shot learning, due to the scarcity of training data for new classes. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Incremental few-shot learning [47, 41, 60, 9, 8] aims to incrementally learn from very few samples. TOPCI [47] proposes a neural gas network to learn and preserve the topology of the feature manifold formed by different classes. FSLL [34] only selects few model parameters for incremental learning and ensures the parameters are close to the optimal ones. To overcome catastrophic forgetting, IDLVQC [8] imposes constraints on the saved exemplars of each class by restricting the embedding drift, and Zhang et al. [59] propose to fix the embedding network for incremental learning. Similar to the finding of Zhang et al., we also discover that an intransigent model that simply does not adapt to new tasks can outperform prior state-of-the-art methods. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Robust optimization. It has been found that flat local minima leads to better generalization capabilities than sharp minima in the sense that a flat minimizer is more robust when the test loss is shifted due to random perturbations [18, 17, 24]. A substantial body of methods [2, 37, 11, 15] have been proposed to optimize neural networks towards flat local minima. In this paper, we show that for incremental few-shot learning, finding flat minima in the base session and tuning the model within the flat region on new tasks can significantly mitigate catastrophic forgetting. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Severity of Catastrophic Forgetting in Incremental Few-Shot Learning ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            171,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Problem Statement ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Incremental few-shot learning (IFL) aims to continually learn to recognize new classes with only few examples. Similar to incremental learning $\\left( \\operatorname { I L } \\right)$ , an IFL model is trained by a sequence of training sessions $\\{ \\mathcal { D } ^ { 1 } , \\cdots , \\mathcal { D } ^ { t } \\}$ , where $\\mathcal { D } ^ { t } = \\{ z _ { i } = ( x _ { i } ^ { t } , y _ { i } ^ { t } ) \\} _ { i }$ is the training data of session $t$ and $\\ v x _ { i } ^ { t }$ is an example of class $y _ { i } ^ { t } \\in \\mathcal { C } ^ { t }$ (the class set of session $t$ ). In IFL, the base session $\\mathcal { D } ^ { 1 }$ usually contains a large number of classes with sufficient training data for each class, while the following sessions $\\left( t \\geq 2 \\right)$ ) only have a small number of classes with few training samples per class, e.g., $\\bar { \\mathcal { D } } ^ { t }$ is often presented as an $N$ -way $K$ -shot task with small $N$ and $K$ . The key difference between $\\mathrm { I L }$ and IFL is, for $\\mathrm { I L }$ , sufficient training data is provided in each session. Similar to $\\mathrm { I L }$ , in each training session $t$ of IFL, the model has only access to the training data $\\mathcal { D } ^ { t }$ and possibly a small amount of saved exemplars from previous sessions. When the training of session $t$ is completed, the model is evaluated on test samples from all encountered classes ${ \\mathcal { C } } = \\textstyle \\bigcup _ { i = 1 } ^ { t } { \\mathcal { C } } ^ { i }$ , where it is assumed that there is no overlap between the classes of different sessions, i.e., $\\forall i , j$ and $i \\neq j$ , ${ \\mathcal { C } } ^ { i } \\cap { \\mathcal { C } } ^ { j } = \\emptyset$ . ",
+        "bbox": [
+            173,
+            233,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Catastrophic forgetting. IFL is undoubtedly a more challenging problem than $\\mathrm { I L }$ due to the data scarcity setting. IL suffers from catastrophic forgetting, a well-known phenomenon and long-standing issue, which refers to the drastic drop in test performance on previous (old) classes, caused by the inaccessibility of old data in the current training session. Unfortunately, catastrophic forgetting is an even bigger issue for IFL, because data scarcity makes it difficult to adapt well to new tasks and learn new concepts, while the adaptation process could easily lead to the forgetting of base classes. In the following, we illustrate this point by evaluating a simple baseline model for IFL. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 A Simple Baseline Model for IFL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            527,
+            441,
+            542
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider an intransigent model that simply does not adapt to new tasks. Particularly, the model only needs to be trained in the base session $\\bar { \\mathcal { D } } ^ { 1 }$ and is directly used for inference in all sessions. ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Training $( t = 1$ ). We train a feature extractor $f$ parameterized by $\\phi$ with a fully-connected layer as classifier by minimizing the standard cross-entropy loss using the training examples of $\\mathcal { D } ^ { 1 }$ . The feature extractor $f$ is fixed for the following sessions $\\left( t \\geq 2 \\right)$ ) without any fine-tuning on new classes. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Inference (test). In each session, the inference is conducted by a simple nearest class mean (NCM) classification algorithm [36]. Specifically, all the training and test samples are mapped to the embedding space of the feature extractor $f$ , and Euclidean distance $d ( \\cdot , \\cdot )$ is used to measure the similarity between them. The classifier is given by ",
+        "bbox": [
+            174,
+            637,
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+            694
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/70bb73688b8fe489dd1aa8a5c9a04f2e856743c6d0b79e46945ef624058d89b6.jpg",
+        "text": "$$\nc _ { k } ^ { \\star } = \\operatorname * { a r g m i n } _ { c \\in \\mathcal { C } } d ( f ( x ; \\phi ) , p _ { c } ) , \\mathrm { ~ w h e r e ~ } p _ { c } = \\frac { 1 } { N _ { c } } \\sum _ { i } \\mathbb { 1 } ( y _ { i } = c ) f ( x _ { i } ; \\phi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            271,
+            704,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { C }$ denotes all the encountered classes, $p _ { c }$ refers to the prototype of class $c$ (the mean vector of all the training samples of class $c$ in the embedding space), and $N _ { c }$ denotes the number of the training images of class $c$ . Note that we save the prototypes of all classes in ${ \\mathcal { C } } ^ { t }$ for later evaluation. ",
+        "bbox": [
+            174,
+            752,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The baseline model outperforms state-of-the-art IFL and IL methods. We compare the above baseline model against state-of-the-art IFL methods including FSLL [34], IDLVQC [8] and TOPIC [47], IL methods including Rebalance [20] and iCarl [40], and a joint-training method that uses all previously seen data including the base and the following few-shot tasks for training, for IFL. The performance is evaluated on miniImageNet, CIFAR-100, and CUB-200. We tune the methods re-implemented by us to the best performance. For the other methods, we use the results reported in the original papers. The experimental details are provided in Sec. 5. As shown in Fig. 1, the baseline model consistently outperforms all the compared methods including the joint-training method (which suffers from severe data imbalance) on every dataset3. The fact that an intransigent model performs best suggests that ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a3479acb9b05770da7d6c9d886cf4da1b35bd210974b3e33e48344c785552875.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of the proposed baseline model with state-of-the-art IFL and $\\mathrm { I L }$ methods and the joint-training method.The baseline model outperforms all the other methods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• For IFL, preserving the old (base classes) may be more critical than adapting to the new. Due to data scarcity, the performance gain on new classes is limited and cannot make up for the significant performance drop on base classes. • Prior works [47, 8, 34, 20, 40] that enforce strong constraints on model parameters or exemplars during fine-tuning on new classes cannot effectively prevent catastrophic forgetting in IFL, indicating that actions may need to be taken in the base training stage. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Overcoming Catastrophic Forgetting in IFL by Finding Flat Minima ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The goal of IFL is to preserve the old while adapting to the new efficiently. The results and analysis in Sec. 3 suggest that it might be “a bit late” to try to prevent catastrophic forgetting in the few-shot learning sessions $\\left( t \\geq 2 \\right)$ ), which motivates us to consider this problem in the base training session. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Overview of our approach. To overcome catastrophic forgetting in IFL, we propose to find a $b$ -flat $( b > 0$ ) local minima $\\theta ^ { \\star }$ of the base training objective function and then fine-tune the model within the flat region in later few-shot learning sessions. Specifically, for any parameter vector $\\theta$ in the flat region, i.e., $\\theta ^ { \\star } - b \\preceq \\theta \\preceq \\theta ^ { \\star } + b$ , the risk (loss) of the base classes is minimized such that the classes are well separated in the embedding space of $f _ { \\theta }$ . In the later incremental few-shot learning sessions $\\left( t \\geq 2 \\right)$ ), we fine-tune the model parameters within this region to learn new classes, i.e., to find ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/758a474c9172aeff13667cc308644ad00c4565c26fbc1968815a0123a8bcbf50.jpg",
+        "text": "$$\n\\theta ^ { \\prime } = \\arg \\operatorname* { m i n } _ { \\theta } \\sum _ { z \\in \\mathcal { D } ^ { t } } \\mathcal { L } ( z ; \\theta ) , \\mathrm { s . t . } \\theta ^ { \\star } - b \\preceq \\theta \\preceq \\theta ^ { \\star } + b .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As such, the fine-tuned model $\\theta ^ { \\prime }$ can adapt to new classes while preserving the old ones. Also, due to the nature of few-shot learning, to avoid excessive training and overfitting, it suffices to tune the model in a relatively small region. A graphical illustration of our approach and prior arts, as well as the notions of sharp minima and flat minima, are presented in Fig. 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Searching for Flat Local Minima in the Base Training Stage ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A formal definition of $b$ -flat local minima is given as follows. ",
+        "bbox": [
+            173,
+            768,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 ( $b$ -Flat Local Minima). Given a real-valued objective function $\\mathcal { L } ( z ; \\theta )$ , for any $b > 0$ , $\\theta ^ { \\star }$ is a $b$ -flat local minima of $\\mathcal { L } ( z ; \\theta )$ , if the following conditions are satisfied. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Condition 1: $\\mathcal { L } ( z ; \\theta ^ { \\star } ) = \\mathcal { L } ( z ; \\theta ^ { \\star } + \\epsilon )$ , where $- \\mathbf { b } \\preceq \\epsilon \\preceq \\mathbf { b }$ and $\\mathbf { b } _ { i } = b$ . ",
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+    {
+        "type": "text",
+        "text": "• Condition 2: there exist $\\mathbf { c } _ { 1 } \\prec \\theta ^ { \\star } - b$ and ${ \\bf c } _ { 2 } \\succ \\theta ^ { \\star } + b ,$ , s.t. $\\mathcal { L } ( z ; \\theta ) > \\mathcal { L } ( z ; \\theta ^ { \\star } )$ , where $\\mathbf { c } _ { 1 } \\prec \\theta \\prec \\theta ^ { \\star } - b$ and $\\mathcal { L } ( z ; \\theta ^ { \\star } ) < \\mathcal { L } ( z ; \\theta )$ , where $\\theta ^ { \\star } + b \\prec \\theta \\prec \\mathbf { c } _ { 2 }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/4aa02a309c91762f668ef9d87403ca2452fa0984b7055d381e241148d4136b17.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of our approach and existing solutions. indicates the incremental learning steps on new classes. $R _ { 1 }$ and $R _ { 2 }$ respectively denote the loss of base classes before and after minimizing the loss of new classes. (a) SGD finds sharp minima in the base training. Directly tuning the model on new classes will result in a severe performance drop on base classes. (b) Enforcing strong constraints on parameters by penalizing parameter changes [1, 28, 34] may still lead to a significant performance drop on base classes. (c) Finding flat local minima of base classes and clamping the parameters after trained on new classes to make them fall within the flat region can effectively mitigate catastrophic forgetting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In practice, it is hard to find the flat local minima that strictly satisfies the above definition, which may not even exist. Hence, our goal is to find an approximately flat local minima of the base training objective function. To this end, we propose to add some small random noise to the model parameters. The noise can be added for multiple times to obtain similar but different loss functions, which will be optimized together to locate the flat minima region. The intuition is clear – the parameter vectors around the flat local minima also have small function values. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To formally state the idea, we assume that the model is parameterized by $\\theta = \\{ \\phi , \\psi \\}$ , where $\\phi$ denotes the parameters of the embedding network and $\\psi$ denotes the parameters of the classifier. $z$ denotes a labelled training sample. Denote the loss function by $\\mathcal { L } \\colon \\dot { \\mathbb { R } } ^ { d _ { z } }  \\mathbb { R }$ . Our target is to minimize the expected loss function $R$ $\\ ? \\colon \\mathbb { R } ^ { d }  \\mathbb { R }$ w.r.t. the joint distribution of data $z$ and noise $\\epsilon$ , i.e., ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/b3a756bf9a16e8cc90d5c6fa60b915c0691b3ef26adf8cd52a8a008429be1de2.jpg",
+        "text": "$$\nR ( \\theta ) = \\int _ { \\mathbb { R } ^ { d _ { \\epsilon } } } \\int _ { \\mathbb { R } ^ { d _ { z } } } \\mathcal { L } ( z ; \\phi + \\epsilon , \\psi ) d P ( z ) d P ( \\epsilon ) = \\mathbb { E } [ \\mathcal { L } ( z ; \\phi + \\epsilon , \\psi ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            541,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $P ( z )$ is the data distribution and $P ( \\epsilon )$ is the noise distribution, and $z$ and $\\epsilon$ are independent. Since it is impossible to minimize the expected loss, we minimize its estimation, the empirical loss, which is given by ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c9d538eacc0da614704f9f7afd89c148c7c17ddb6eaf97d6db3174866778fcfe.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta ) = \\frac { 1 } { M } \\sum _ { j = 1 } ^ { M } \\mathcal { L } _ { \\mathrm { b a s e } } ( z ; \\phi + \\epsilon _ { j } , \\psi ) , \\mathrm { w h e r e }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            616,
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+            661
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05bd6bf14baee9feb9e7e27de4100c40dd1dd920a7b2b2c1d45352a0ed68f0ae.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { b a s e } } ( z ; \\phi + \\epsilon _ { j } , \\psi ) = \\frac { 1 } { | \\mathcal { D } ^ { 1 } | } \\sum _ { z \\in \\mathcal { D } ^ { 1 } } \\mathcal { L } _ { c e } ( z ; \\phi + \\epsilon _ { j } , \\psi ) + \\lambda \\frac { 1 } { | \\mathcal { C } ^ { 1 } | } \\sum _ { c \\in \\mathcal { C } ^ { 1 } } \\| p _ { c } - p _ { c } ^ { * } \\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            664,
+            750,
+            702
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon _ { j }$ is a noise vector sampled from $P ( \\epsilon )$ , $M$ is the sampling times, $\\mathcal { L } _ { c e } ( z ; \\phi + \\epsilon _ { j } , \\psi )$ refers to the cross-entropy loss of a training sample $z$ , and $p _ { c }$ and $p _ { c } ^ { * }$ are the class prototypes before and after injecting noise respectively. The first term of $\\mathcal { L } _ { b a s e }$ is designed to find the flat region where the parameters $\\phi$ of the embedding network can well separate the base classes. The second term enforces the class prototypes fixed within such region, which is designed to solve the prototype drift problem [54, 8] (the class prototypes change after updating the network) in later incremental learning sessions such that the saved base class prototypes can be directly used for evaluation in later sessions. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Incremental Few-shot Learning within the Flat Region ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the incremental few-shot learning sessions $\\left( t \\geq 2 \\right)$ ), we fine-tune the parameters $\\phi$ of the embedding network within the flat region to learn new classes. It is worth noting that while the flat region might be relatively small, it is enough for incremental few-shot learning. Because only few training samples are provided for each new class, to prevent overfitting in few-shot learning, excessive training should be avoided and only a small number of update iterations should be applied. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: F2M ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: the flat region bound $b$ , randomly initialized $\\theta = \\{ \\phi , \\psi \\}$ , the step sizes $\\alpha$ and $\\beta$ .   \n// Training over base classes $t = 1$   \nfor epoch $k = 1 , 2 , \\dots$ do for $j = 1 , 2 , . . . , M$ do Sample a noise vector $\\epsilon _ { j } \\sim P ( \\epsilon )$ , s.t. $- \\mathbf { b } \\preceq \\epsilon _ { j } \\preceq \\mathbf { b }$ ; Add the noise to the parameters of the embedding network, i.e., $\\theta = \\{ \\phi + \\epsilon _ { j } , \\psi \\}$ ; Compute the base loss $\\mathcal { L } _ { b a s e }$ with Eq. 4; Reset the parameters, i.e., $\\theta = \\{ \\phi , \\psi \\}$ ; end Update $\\theta = \\theta - \\alpha \\nabla { \\mathcal { L } } ( \\theta )$ with the loss $\\mathcal { L }$ defined in Eq. 3.   \nend   \nNormalize and save the prototype of each base class;   \n// Incremental learning $t \\geq 2$   \nCombine the training data $\\mathcal { D } ^ { t }$ and the exemplars saved in previous few-shot sessions $2 \\leq t _ { e } < t$ ;   \nfor epoch $k = 1 , 2 , . .$ . do Compute the metric-based classification loss ${ \\mathcal { L } } _ { m }$ by Eq. 5; Update $\\phi = \\phi - \\beta \\nabla \\mathcal { L } _ { m } ( z ; \\phi )$ ; Clamp the parameters $\\phi$ to ensure they fall in the flat minima region;   \nend   \nRandomly select and save a few exemplars from the training data $\\mathcal { D } ^ { t }$ ;   \nNormalize and save the prototype of each new class;   \nOutput: Model parameters $\\theta = \\{ \\phi , \\psi \\}$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We employ a metric-based classification algorithm with Euclidean distance to fine-tune the parameters. The loss function is defined as ",
+        "bbox": [
+            173,
+            457,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cbd0eea1d58f8b6facc442c6b0aaff2ab86faf776744b94665336624169f0ba3.jpg",
+        "text": "$$\n\\mathcal { L } _ { m } ( z ; \\phi ) = - \\sum _ { z \\in \\mathcal { D } } \\sum _ { c \\in \\mathcal { C } } \\mathbb { 1 } ( y = c ) \\log \\big ( \\frac { e ^ { - d ( p _ { c } , f ( x ; \\phi ) ) } } { \\sum _ { c _ { k } \\in \\mathcal { C } } e ^ { - d ( p _ { c _ { k } } , f ( x ; \\phi ) ) } } \\big ) , z\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            484,
+            709,
+            526
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $d ( \\cdot , \\cdot )$ denotes Euclidean distance, $p _ { c }$ is the prototype of class $c$ , $\\textstyle { \\mathcal { C } } = \\bigcup _ { i = 1 } ^ { t } { \\mathcal { C } } ^ { i }$ refers to all encountered classes, and ${ \\mathcal { D } } = { \\mathcal { D } } ^ { t } \\cup { \\mathcal { P } }$ denotes the union of the current training data $\\mathcal { D } ^ { t }$ and the exemplar set $\\mathcal { P } = \\{ P _ { 2 } , . . . , P _ { t - 1 } \\}$ , where $P _ { t _ { e } } ( 2 \\leq t _ { e } < t )$ is the set of saved exemplars in session $t _ { e }$ . Note that the prototypes of new classes are computed by Eq. 1, and those of base classes are saved in the base session. After updating the embedding network parameters, we clamp them to ensure that they fall within the flat region, i.e. $\\phi ^ { \\star } - b \\preceq \\phi \\preceq \\phi ^ { \\star } + b$ , where $\\phi ^ { \\star }$ denotes the optimal parameter vector learned in the base session. After fine-tuning, we evaluate the model using the nearest class mean classifier as in Eq. 1, with previously saved prototypes and newly computed ones. The whole training process is described in Algorithm 1. Note that to calibrate the estimates of the classifier, we normalize all prototypes to make those of base classes and those of new classes have the same norm. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Convergence Analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our aim is to find a flat region within which all parameter vectors work well. We then minimize the expected loss w.r.t. the joint distribution of noise $\\epsilon$ and data $z$ . To approximate this expected loss, we sample from $P ( \\epsilon )$ for multiple times in each iteration and optimize the objective function using stochastic gradient descent (SGD). Here, we provide theoretical guarantees for our method. Given the non-convex loss function in Eq. 3, we prove the convergence of our proposed method. The proof idea is inspired by the convergence analysis of SGD [3, 27]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Formally, in each batch $k$ , let $z _ { k }$ denote the batch data, $\\{ \\epsilon _ { j } \\} _ { j = 1 } ^ { M }$ be the sampled noises, and $\\alpha _ { k }$ be the step size. In the base training session, we update the model parameters as follows: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f6c9a3d87022e40ce81b39a403fe4cc48a58bc675e9f86e603ac16bde127748e.jpg",
+        "text": "$$\n\\theta _ { k + 1 } = \\theta _ { k } - \\frac { \\alpha _ { k } } { M } \\sum _ { j = 1 } ^ { M } \\nabla \\mathcal { L } _ { \\mathrm { b a s e } } \\big ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } \\big ) = \\theta _ { k } - \\frac { \\alpha _ { k } } { M } \\sum _ { j = 1 } ^ { M } g \\big ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            829,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } ) = \\nabla { \\mathcal { L } } _ { \\mathrm { b a s e } } ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } )$ is the gradient. To formally analyze the convergence of our algorithm, we define the following assumptions. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.1 (L-smooth risk function). The expected loss function $R : \\mathbb { R } ^ { d }  \\mathbb { R }$ (Eq. 2) is continuously differentiable and $L$ -smooth with constant $L > 0$ such that ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c79ae89ed0149ed19421c38fca4ad7578dc6b6f98eb0b23f11ea9772ff205480.jpg",
+        "text": "$$\n\\| \\nabla R ( \\theta ) - \\nabla R ( \\theta ^ { \\prime } ) \\| _ { 2 } \\leq L \\| \\theta - \\theta ^ { \\prime } \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            126,
+            617,
+            143
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This assumption is significant for the convergence analysis of gradient-based optimization algorithms, since it limits how fast the gradient of the loss function can change w.r.t. the parameter vector. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.2. The expected loss function satisfies the following conditions: ",
+        "bbox": [
+            176,
+            189,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Condition 1: $R$ is bounded below by a scalar $R ^ { \\star }$ , given the sequence of parameters $\\{ \\theta _ { k } \\}$ ",
+        "bbox": [
+            209,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Condition 2: For all $k \\in \\mathbb N$ and $j \\in [ 1 , M ]$ , ",
+        "bbox": [
+            218,
+            238,
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+            255
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5bc16b0b8564721592171d4aa24ec24ad8a4402750b2216466dfbe55ca25b01d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { z _ { k } , \\epsilon _ { j } } \\big [ g \\big ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } \\big ) \\big ] = \\nabla R ( \\theta _ { k } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            260,
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+            279
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Condition $^ 3$ : There exist scalars $m _ { 1 } \\geq 0$ and $m _ { 2 } \\geq 0 ,$ , for all $k \\in \\mathbb N$ and $j \\in [ 1 , M ]$ , ",
+        "bbox": [
+            217,
+            290,
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+            308
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5096183562af805022f33c150d89bccb14263ee7a806f9ee4921f24c132bfe54.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { V } _ { z _ { k } , \\epsilon _ { j } } [ g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } ) ] \\le m _ { 1 } + m _ { 2 } \\| \\nabla R ( \\theta _ { k } ) \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            313,
+            699,
+            333
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbb { E } _ { z _ { k } , \\epsilon _ { j } } [ \\cdot ]$ denotes the expectation w.r.t. the joint distribution of random variables $z _ { k }$ and $\\epsilon _ { j }$ , and $\\mathbb { V } _ { z _ { k } , \\epsilon _ { j } } [ \\cdot ]$ denotes the variance. Condition 1 ensures that the expected loss $R$ is bounded by a minimum value $R ^ { \\star }$ during the updates, which is a natural and practical assumption. Condition 2 assumes that the gradient $g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } )$ is an unbiased estimate of $\\nabla R ( \\theta _ { k } )$ . This is a strict assumption made to simplify the proof, but it can be easily relaxed to a general and easily-met condition that there exist $\\mu _ { 1 } \\geq \\mu _ { 2 } > 0$ satisfying $\\| \\mathbb { E } _ { z _ { k } , \\epsilon _ { j } } [ \\bar { g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } ) } ] \\| _ { 2 } \\leq \\mu _ { 1 } \\| \\nabla \\bar { R ( \\theta _ { k } ) } \\| _ { 2 }$ and $\\nabla R ( \\theta _ { k } ) ^ { T } \\mathbb { E } _ { z _ { k } , \\epsilon _ { j } } [ g ( \\underline { { z _ { k } } } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } ) ] \\geq \\mu _ { 2 } \\| \\nabla R ( \\theta _ { k } ) \\| _ { 2 } ^ { 2 }$ . Therefore, the convergence can be proved in a similar way using the techniques presented in the Appendix. Condition 3 assumes the variance of the gradient $g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } )$ cannot be arbitrarily large, which is also reasonable in practice. To facilitate later analysis, similar to [43], we restrict the step sizes as follows. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.3. The learning rates satisfy: ",
+        "bbox": [
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+            492,
+            467,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11b47e921b3b0cb564364a1725626d441628fc2d3166e3268b98c43dcd1fe07f.jpg",
+        "text": "$$\n\\sum _ { k = 1 } ^ { \\infty } \\alpha _ { k } = \\infty , \\ \\sum _ { k = 1 } ^ { \\infty } \\alpha _ { k } ^ { 2 } < \\infty .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This assumption can be easily met, since in practice the learning rate $\\alpha _ { k }$ is usually far less than 1 and decreases w.r.t. $k$ . Based on the above assumptions, we can derive the following theorem. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1. Under assumptions 4.1, 4.2 and 4.3, we further assume that the risk function $R$ is twice differentiable, and that $\\| \\nabla R ( \\theta ) \\| _ { 2 } ^ { 2 }$ is $L _ { 2 }$ -smooth with constant $L _ { 2 } > 0$ , then we have ",
+        "bbox": [
+            173,
+            599,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/19bd76fc1ebc8d0c84ad22f6f47d9db08d8dec6c80681989c406d1873676a886.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { k \\to \\infty } \\mathbb { E } [ \\| \\nabla R ( \\theta _ { k } ) \\| _ { 2 } ^ { 2 } ] = 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            635,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This theorem establishes the convergence of our algorithm. The proof is provided in Appendix A.1. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we empirically evaluate our proposed method for incremental few-shot learning and demonstrate its effectiveness by comparison with state-of-the-art methods. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Experimental Setup ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets. For CIFAR-100 and miniImageNet, we randomly select 60 classes as the base classes and the remaining 40 classes as the new classes. In each incremental learning session, we construct 5-way 5-shot tasks by randomly picking 5 classes and sampling 5 examples for each class. For CUB-200-2011 with 200 classes, we select 100 classes as the base classes and 100 classes as the new ones. We test 10-way 5-shot tasks on this dataset. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baselines. We compare our method F2M with 8 methods: the Baseline proposed in Sec. 3, a jointtraining method that uses all previously seen data including the base and the following few-shot tasks for training, the classifier re-training method (cRT) [25] for long-tailed classification trained with all encountered data, iCaRL [40], Rebalance [20], TOPIC [47], FSLL [34], and IDLVQ-C [8]. For a fair comparison, we re-implement cRT [25], iCaRL [40], Rebalance [20], FSLL [34], and the joint-training method and tune them to their best performance. We also provide the results reported in the original papers for comparison. The results of TOPIC [47] and IDLVQ-C [8] are copied from the original papers. Note that for IL, joint-training is naturally the upper bound of incremental learning algorithms, however, for IFL, joint-training is not a good approximation of the upper bound because data imbalance makes the model perform significantly poorer on new classes (long-tailed classes). To address the data imbalance issue, we re-implement the cRT method as the approximate upper bound. ",
+        "bbox": [
+            173,
+            882,
+            821,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/3835b07fcacb259d9fc8f7adfa333925ec8fddfe46906fed2c496f5cf8b28580.jpg",
+        "table_caption": [
+            "Table 1: Classification accuracy on CIFAR-100 for 5-way 5-shot incremental learning. ∗ indicates our re-implementation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"9\">sessions</td><td rowspan=\"2\">The gap with cRT</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>cRT[25]*</td><td>65.18</td><td>63.89</td><td>60.20</td><td>57.23</td><td>53.71</td><td>50.39</td><td>48.77</td><td>47.29</td><td>45.28</td><td>=</td></tr><tr><td>Joint-training*</td><td>65.18</td><td>61.45</td><td>57.36</td><td>53.68</td><td>50.84</td><td>47.33</td><td>44.79</td><td>42.62</td><td>40.08</td><td>-5.20</td></tr><tr><td>Baseline</td><td>65.18</td><td>61.67</td><td>58.61</td><td>55.11</td><td>51.86</td><td>49.43</td><td>47.60</td><td>45.64</td><td>43.83</td><td>-1.45</td></tr><tr><td>iCaRL[40]*</td><td>66.52</td><td>57.26</td><td>54.27</td><td>50.62</td><td>47.33</td><td>44.99</td><td>43.14</td><td>41.16</td><td>39.49</td><td>-5.79</td></tr><tr><td>Rebalance [20]*</td><td>66.66</td><td>61.42</td><td>57.29</td><td>53.02</td><td>48.85</td><td>45.68</td><td>43.06</td><td>40.56</td><td>38.35</td><td>-6.93</td></tr><tr><td>FSLL [34]*</td><td>65.18</td><td>56.24</td><td>54.55</td><td>51.61</td><td>49.11</td><td>47.27</td><td>45.35</td><td>43.95</td><td>42.22</td><td>-3.08</td></tr><tr><td>iCaRL[40]</td><td>64.10</td><td>53.28</td><td>41.69</td><td>34.13</td><td>27.93</td><td>25.06</td><td>20.41</td><td>15.48</td><td>13.73</td><td>-31.55</td></tr><tr><td>Rebalance [20]</td><td>64.10</td><td>53.05</td><td>43.96</td><td>36.97</td><td>31.61</td><td>26.73</td><td>21.23</td><td>16.78</td><td>13.54</td><td>-31.74</td></tr><tr><td>TOPIC [47]</td><td>64.10</td><td>55.88</td><td>47.07</td><td>45.16</td><td>40.11</td><td>36.38</td><td>33.96</td><td>31.55</td><td>29.37</td><td>-15.91</td></tr><tr><td>FSLL [34]</td><td>64.10</td><td>55.85</td><td>51.71</td><td>48.59</td><td>45.34</td><td>43.25</td><td>41.52</td><td>39.81</td><td>38.16</td><td>-7.12</td></tr><tr><td>FSLL+SS [34]</td><td>66.76</td><td>55.52</td><td>52.20</td><td>49.17</td><td>46.23</td><td>44.64</td><td>43.07</td><td>41.20</td><td>39.57</td><td>-5.71</td></tr><tr><td>F2M</td><td>64.71</td><td>62.05</td><td>59.01</td><td>55.58</td><td>52.55</td><td>49.96</td><td>48.08</td><td>46.28</td><td>44.67</td><td>-0.61</td></tr></table>",
+        "bbox": [
+            209,
+            125,
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+            309
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a4fbc03180e63e236407b6f9ed5fcce1cee014bd347b0f0a3ce14d1440d31067.jpg",
+        "table_caption": [
+            "Table 2: Classification accuracy on miniImageNet for 5-way 5-shot incremental learning. ∗ indicates our re-implementation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"9\">sessions</td><td rowspan=\"2\">The gap with cRT</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>cRT[25]*</td><td>67.30</td><td>64.15</td><td>60.59</td><td>57.32</td><td>54.22</td><td>51.43</td><td>48.92</td><td>46.78</td><td>44.85</td><td>=</td></tr><tr><td>Joint-training*</td><td>67.30</td><td>62.34</td><td>57.79</td><td>54.08</td><td>50.93</td><td>47.65</td><td>44.64</td><td>42.61</td><td>40.29</td><td>-4.56</td></tr><tr><td>Baseline</td><td>67.30</td><td>63.18</td><td>59.62</td><td>56.33</td><td>53.28</td><td>50.50</td><td>47.96</td><td>45.85</td><td>43.88</td><td>-0.97</td></tr><tr><td>iCaRL [40]*</td><td>67.35</td><td>59.91</td><td>55.64</td><td>52.60</td><td>49.43</td><td>46.73</td><td>44.13</td><td>42.17</td><td>40.29</td><td>-4.56</td></tr><tr><td>Rebalance [20]*</td><td>67.91</td><td>63.11</td><td>58.75</td><td>54.83</td><td>50.68</td><td>47.11</td><td>43.88</td><td>41.19</td><td>38.72</td><td>-6.13</td></tr><tr><td>FSLL [34]*</td><td>67.30</td><td>59.81</td><td>57.26</td><td>54.57</td><td>52.05</td><td>49.42</td><td>46.95</td><td>44.94</td><td>42.87</td><td>-1.11</td></tr><tr><td>iCaRL[40]</td><td>61.31</td><td>46.32</td><td>42.94</td><td>37.63</td><td>30.49</td><td>24.00</td><td>20.89</td><td>18.80</td><td>17.21</td><td>-27.64</td></tr><tr><td>Rebalance [20]</td><td>61.31</td><td>47.80</td><td>39.31</td><td>31.91</td><td>25.68</td><td>21.35</td><td>18.67</td><td>17.24</td><td>14.17</td><td>-30.68</td></tr><tr><td>TOPIC [47]</td><td>61.31</td><td>50.09</td><td>45.17</td><td>41.16</td><td>37.48</td><td>35.52</td><td>32.19</td><td>29.46</td><td>24.42</td><td>-20.43</td></tr><tr><td>FSLL [34]</td><td>66.48</td><td>61.75</td><td>58.16</td><td>54.16</td><td>51.10</td><td>48.53</td><td>46.54</td><td>44.20</td><td>42.28</td><td>-2.57</td></tr><tr><td>FSLL+SS[34]</td><td>68.85</td><td>63.14</td><td>59.24</td><td>55.23</td><td>52.24</td><td>49.65</td><td>47.74</td><td>45.23</td><td>43.92</td><td>-0.93</td></tr><tr><td>IDLVQ-C[8]</td><td>64.77</td><td>59.87</td><td>55.93</td><td>52.62</td><td>49.88</td><td>47.55</td><td>44.83</td><td>43.14</td><td>41.84</td><td>-3.01</td></tr><tr><td>F2M</td><td>67.28</td><td>63.80</td><td>60.38</td><td>57.06</td><td>54.08</td><td>51.39</td><td>48.82</td><td>46.58</td><td>44.65</td><td>-0.20</td></tr></table>",
+        "bbox": [
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+            382,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            627,
+            825,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Experimental details. The experiments are conducted with NVIDIA GPU RTX3090 on CUDA 11.0. We randomly split each dataset into multiple tasks (sessions). For each dataset (with a fixed split), we run each algorithm for 10 times and report the mean accuracy. We adopt ResNet18 [16] as the backbone network. For data augmentation, we use standard random crop and horizontal flip. In the base training stage, we select the last 4 or 8 convolution layers to inject noise, because these layers output higher-level feature representations. The flat region bound $b$ is set as 0.01. We set the number of times for noise sampling as $M = 2 \\sim 4$ , since a larger $M$ will increase the training time. In each incremental few-shot learning session, the total number of training epochs is 6, and the learning rate is 0.02. To verify the correctness of our implementation, we conduct experiments on incremental learning and compare our results to those reported on CIFAR-100 in Appendix A.3. More experiment details are provided in Appendix A.2. ",
+        "bbox": [
+            174,
+            758,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/60bfafebee6bfe1f989d8626c31f85d26fba947b336f7e74dd693733a3e8bad9.jpg",
+        "table_caption": [
+            "Table 3: Classification accuracy on CUB-200-2011 for 10-way 5-shot incremental learning.∗ indicates our re-implementation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"10\">sessions</td><td rowspan=\"2\">The gap with cRT</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>cRT[25]*</td><td>80.83</td><td>78.51</td><td>76.12</td><td>73.93</td><td>71.46</td><td>68.96</td><td>67.73</td><td>66.75</td><td>64.22</td><td>62.53</td><td>61.08</td><td></td></tr><tr><td>Joint-training*</td><td>80.83</td><td>77.57</td><td>74.11</td><td>70.75</td><td>68.52</td><td>65.97</td><td>64.58</td><td>62.22</td><td>60.18</td><td>58.49</td><td>56.78</td><td>-4.30</td></tr><tr><td>Baseline</td><td>80.87</td><td>77.15</td><td>74.46</td><td>72.26</td><td>69.47</td><td>67.18</td><td>65.62</td><td>63.68</td><td>61.30</td><td>59.72</td><td>58.12</td><td>-2.96</td></tr><tr><td>iCaRL [40]*</td><td>79.58</td><td>67.63</td><td>64.17</td><td>61.80</td><td>58.10</td><td>55.51</td><td>53.34</td><td>50.89</td><td>48.62</td><td>47.34</td><td>45.60</td><td>-15.48</td></tr><tr><td>Rebalance [20]*</td><td>80.94</td><td>70.32</td><td>62.96</td><td>57.19</td><td>51.06</td><td>46.70</td><td>44.03</td><td>40.15</td><td>36.75</td><td>34.88</td><td>32.09</td><td>-28.99</td></tr><tr><td>FSLL [34]*</td><td>80.83</td><td>77.38</td><td>72.37</td><td>71.84</td><td>67.51</td><td>65.30</td><td>63.75</td><td>61.16</td><td>59.05</td><td>58.03</td><td>55.82</td><td>-5.26</td></tr><tr><td>iCaRL [40]</td><td>68.68</td><td>52.65</td><td>48.61</td><td>44.16</td><td>36.62</td><td>29.52</td><td>27.83</td><td>26.26</td><td>24.01</td><td>23.89</td><td>21.16</td><td>-39.92</td></tr><tr><td>Rebalance [20]</td><td>68.68</td><td>57.12</td><td>44.21</td><td>28.78</td><td>26.71</td><td>25.66</td><td>24.62</td><td>21.52</td><td>20.12</td><td>20.06</td><td>19.87</td><td>-41.21</td></tr><tr><td>TOPIC[47]</td><td>68.68</td><td>62.49</td><td>54.81</td><td>49.99</td><td>45.25</td><td>41.40</td><td>38.35</td><td>35.36</td><td>32.22</td><td>28.31</td><td>26.28</td><td>-34.80</td></tr><tr><td>FSLL [34]</td><td>72.77</td><td>69.33</td><td>65.51</td><td>62.66</td><td>61.10</td><td>58.65</td><td>57.78</td><td>57.26</td><td>55.59</td><td>55.39</td><td>54.21</td><td>-6.87</td></tr><tr><td>FSLL+SS [34]</td><td>75.63</td><td>71.81</td><td>68.16</td><td>64.32</td><td>62.61</td><td>60.10</td><td>58.82</td><td>58.70</td><td>56.45</td><td>56.41</td><td>55.82</td><td>-5.26</td></tr><tr><td>IDLVQ-C [8]</td><td>77.37</td><td>74.72</td><td>70.28</td><td>67.13</td><td>65.34</td><td>63.52</td><td>62.10</td><td>61.54</td><td>59.04</td><td>58.68</td><td>57.81</td><td>-3.27</td></tr><tr><td>F2M</td><td>81.07</td><td>78.16</td><td>75.57</td><td>72.89</td><td>70.86</td><td>68.17</td><td>67.01</td><td>65.26</td><td>63.36</td><td>61.76</td><td>60.26</td><td>-0.82</td></tr></table>",
+        "bbox": [
+            176,
+            125,
+            823,
+            313
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/a77af395b80eaa563bb4d8acb99dd5af27c8760a1d705c17e16f8c7db5a960dc.jpg",
+        "table_caption": [
+            "Table 4: Comparison of the flatness of the local minima found by the Baseline and our F2M. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Indicator I</td><td colspan=\"2\">Variance g2</td></tr><tr><td>Training Set</td><td>Testing Set</td><td>Training Set</td><td>Testing Set</td></tr><tr><td>Baseline</td><td>0.2993</td><td>0.4582</td><td>0.1451</td><td>0.2395</td></tr><tr><td>F2M</td><td>0.0506</td><td>0.0800</td><td>0.0296</td><td>0.0334</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Comparison with the State-of-the-Art ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "F2M outperforms the state-of-the-art methods. The main results on CIFAR-100, miniImageNet and CUB-200-2011 are presented in Table 1, Table 2 and Table 3 respectively. Based on the experiment results, we have the following observations: 1) The Baseline introduced in Sec. 3 outperforms the state-of-the-art approaches on all incremental sessions. 2) As expected, cRT consistently outperforms the Baseline up to $1 \\%$ to $3 \\%$ by considering the data imbalance problem and applying proper techniques to tackle the long-tailed classification problem to improve performance. Hence, it is reasonable to use cRT as the approximate upper bound of IFL. 3) Our F2M outperforms the state-ofthe-art methods and the Baseline. Moreover, the performance of F2M is very close to the approximate upper bound, i.e., the gap with cRT is only $0 . 2 \\%$ in the last session on miniImageNet. The results show that even with strong constraints [20, 40, 34] and saved examplars of base classes [20, 40, 8], current methods cannot effectively address the catastrophic forgetting problem. In contrast, finding flat minima seems a promising approach to overcome this harsh problem. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3 Ablation Study and Analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Analysis on the flatness of local minima. Here, we verify that our method can find a more flat local minima than the Baseline. For a found local minima $\\theta ^ { \\star }$ , we measure its flatness as follows. We sample the noise for 1000 times. For each time, we inject the sampled noise to $\\theta ^ { \\star }$ and calculate the loss $\\mathcal { L } _ { i }$ . to measure the flatness. Then, we adopt the indicator I = 11000 P1000i=1 $\\mathcal { L } ^ { * }$ denotes the loss of $\\begin{array} { r } { I = \\frac { 1 } { 1 0 0 0 } \\sum _ { i = 1 } ^ { 1 0 0 0 } ( \\mathcal { L } _ { i } - \\mathcal { L } ^ { * } ) ^ { 2 } } \\end{array}$ $\\theta ^ { \\star }$ , and $\\overline { { \\mathcal { L } } }$ 1000denotes the average loss of and variance $\\begin{array} { r } { \\sigma ^ { 2 } = \\frac { 1 } { 1 0 0 0 } \\sum _ { i = 1 } ^ { 1 0 0 0 } ( \\mathcal { L } _ { i } - \\overline { { \\mathcal { L } } } ) ^ { 2 } } \\end{array}$ $\\{ \\mathcal { L } _ { i } \\} _ { i = 1 } ^ { 1 0 0 0 }$ . The values of the indicator and variance of F2M and the Baseline are presented in Table 4, which clearly demonstrate that our method can find a more flat local minima. ",
+        "bbox": [
+            173,
+            685,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ablation study on the designs of our method. Here, we study the effectiveness of each design of our method, including adding noise to the model parameters for finding $b$ -flat local minima (FM) during the base training session, the prototype fixing term (PF) used in the base training objective (Eq. 4), parameter clamping (PC) during incremental learning, and prototype normalization (PN). We conduct an ablation study by removing each component in turn and report the experimental results in Table 5. ",
+        "bbox": [
+            174,
+            794,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finding b-flat local minima. Standard supervised training with SGD as the optimizer tends to converge to a sharp local minima. It leads to a significant drop in performance because the loss changes quickly in the neighborhood of the sharp local minima. As shown in Table 5, even with parameter clamping during incremental learning, the performance still drops significantly. In contrast, restricting the parameters in a small flat region can mitigate the forgetting problem. ",
+        "bbox": [
+            174,
+            882,
+            821,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/d977f14d1af3cb4fbf8c66feb0d1ae35cc2c4ff64227c1cc271d7baf8cf1dedb.jpg",
+        "table_caption": [
+            "Table 5: Ablation study of our F2M on CIFAR-100. PD refers to the performance dropping rate. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">FM</td><td rowspan=\"2\">PF</td><td rowspan=\"2\">PC</td><td rowspan=\"2\">PN</td><td colspan=\"10\">sessions</td><td rowspan=\"2\">PD↓</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td rowspan=\"4\">广</td><td></td><td></td><td></td><td>65.18</td><td>60.83</td><td>53.13</td><td>43.57</td><td>23.75</td><td>10.76</td><td>08.26</td><td>07.24</td><td>06.45</td><td>58.73</td></tr><tr><td></td><td>√</td><td></td><td>65.18</td><td>59.48</td><td>56.77</td><td>52.99</td><td>50.09</td><td>47.80</td><td>45.92</td><td>44.20</td><td>42.55</td><td>22.63</td></tr><tr><td>√</td><td></td><td>√</td><td>64.71</td><td>59.54</td><td>53.03</td><td>45.09</td><td>41.68</td><td>39.04</td><td>38.64</td><td>37.19</td><td>36.01</td><td>28.70</td></tr><tr><td></td><td>√</td><td>√</td><td>64.55</td><td>61.27</td><td>58.33</td><td>54.82</td><td>51.60</td><td>49.22</td><td>47.48</td><td>45.78</td><td>44.08</td><td>20.47</td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td>64.71</td><td>61.75</td><td>58.80</td><td>55.33</td><td>52.27</td><td>49.75</td><td>47.72</td><td>46.01</td><td>44.43</td><td>20.28</td></tr><tr><td>√</td><td></td><td>√</td><td>√</td><td>64.71</td><td>61.99</td><td>58.99</td><td>55.58</td><td>52.55</td><td>49.96</td><td>48.08</td><td>46.28</td><td>44.67</td><td>20.04</td></tr></table>",
+        "bbox": [
+            202,
+            111,
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+            223
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/10f2e54af4ac4a6df16159258b656661085cb0924e08a7dd026b9539505353a3.jpg",
+        "table_caption": [
+            "Table 6: Study of the flat region bound $b$ for 5-way 5-shot incremental learning on CIFAR-100. The top 3 results in each row are in boldface. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Session</td><td colspan=\"6\">The hyperparameter b</td></tr><tr><td>0.0025</td><td>0.005</td><td>0.01</td><td>0.02</td><td>0.04</td><td>0.08</td></tr><tr><td>Session 1(60 bases classes)</td><td>64.85</td><td>64.67</td><td>64.81</td><td>64.71</td><td>63.30</td><td>62.25</td></tr><tr><td>Session 9 (All 100 classes)</td><td>44.16</td><td>44.54</td><td>44.58</td><td>44.67</td><td>43.75</td><td>43.04</td></tr><tr><td>Session 9 (60 base classes)</td><td>59.58</td><td>59.69</td><td>59.73</td><td>59.44</td><td>58.38</td><td>57.21</td></tr><tr><td>Session 9 (40 new classes)</td><td>21.03</td><td>21.81</td><td>21.86</td><td>22.52</td><td>21.80</td><td>21.77</td></tr></table>",
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+        "text": "Prototype fixing. Without fixing the prototypes after injecting noise to selected layers during the process of finding local minima, i.e. removing the second term of Eq. 4, it is still possible to tune the model within the flat region to well separate base classes. However, the saved prototypes of base classes will become less accurate because the embeddings of the base samples suffer from semantic drift [54]. As shown in Table 5, it results in a performance drop of nearly $0 . 6 \\%$ . ",
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+        "type": "text",
+        "text": "Parameter clamping. Parameter clamping restricts the model parameters to the $b$ -flat region after incremental few-shot learning. Outside the $b$ -flat region, the performance drops quickly. It can be seen from Table 5 that removing parameter clamping leads to a significant drop in performance. ",
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+        "text": "Prototype normalization. As mentioned in Sec. 4.2, we normalize the class prototypes to calibrate the estimates of the class mean classifier. The results in Table 5 show the effectiveness of normalization, which helps to further improve the performance. ",
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+        "text": "Study of the flat region bound $b$ . We study the effect of the flat region bound $b$ for 5-way 5-shot incremental learning on CIFAR-100. We report the test accuracy in session 1 (base session) and session 9 (last session) w.r.t. different $b$ in Table 6. It can be seen that the best results are achieved for $b \\in [ 0 . 0 0 5 , 0 . 0 2 ]$ . A larger $b$ (e.g., 0.04 or 0.08) leads to a significant performance drop on base classes, even for those in session 1, indicating that there may not exist a large flat region around a good local minima. Meanwhile, a smaller $b$ (e.g., 0.0025) results in a performance decline on new classes, due to the overly small capacity of the flat region. This illustrates the trade-off effect of $b$ . ",
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+        "text": "6 Conclusion ",
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+        "text": "We have proposed a novel approach to overcome catastrophic forgetting in incremental few-shot learning by finding flat local minima of the objective function in the base training stage and then fine-tuning the model within the flat region on new tasks. Extensive experiments on benchmark datasets show that our model can effectively mitigate catastrophic forgetting and adapt to new classes. A limitation of our method is that it may not be suitable for medium- or high-shot tasks, since the flat region is relatively small, which limits the model capacity. However, it is still possible to adapt our core idea for incremental learning. For example, one can search for a less flat but wider local minima region in the base training stage and tune the model within this region during incremental learning sessions, where previous techniques such as elastic weight consolidation (EWC) [28] can be used to constraint the model parameters. This could be an interesting direction for future research. ",
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+        "text": "Acknowledgments and Disclosure of Funding ",
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+        "text": "We would like to thank the anonymous reviewers for their insightful and helpful comments. This research was supported by the grant of DaSAIL project P0030935 funded by PolyU/UGC. ",
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parse/train/BJij4yg0Z/BJij4yg0Z.md

@@ -0,0 +1,326 @@
+# A BAYESIAN PERSPECTIVE ON GENERALIZATION AND STOCHASTIC GRADIENT DESCENT
+
+Samuel L. Smith∗ & Quoc V. Le Google Brain {slsmith, qvl}@google.com
+
+# ABSTRACT
+
+We consider two questions at the heart of machine learning; how can we predict if a minimum will generalize to the test set, and why does stochastic gradient descent find minima that generalize well? Our work responds to Zhang et al. (2016), who showed deep neural networks can easily memorize randomly labeled training data, despite generalizing well on real labels of the same inputs. We show that the same phenomenon occurs in small linear models. These observations are explained by the Bayesian evidence, which penalizes sharp minima but is invariant to model parameterization. We also demonstrate that, when one holds the learning rate fixed, there is an optimum batch size which maximizes the test set accuracy. We propose that the noise introduced by small mini-batches drives the parameters towards minima whose evidence is large. Interpreting stochastic gradient descent as a stochastic differential equation, we identify the “noise scale” $\begin{array} { r } { \overline { { g } } = \epsilon ( \frac { N } { B } - 1 ) \approx } \end{array}$ $\epsilon N / B$ , where $\epsilon$ is the learning rate, $N$ the training set size and $B$ the batch size. Consequently the optimum batch size is proportional to both the learning rate and the size of the training set, $B _ { o p t } \propto \epsilon N$ . We verify these predictions empirically.
+
+# 1 INTRODUCTION
+
+This paper shows Bayesian principles can explain many recent observations in the deep learning literature, while also discovering practical new insights. Zhang et al. (2016) trained deep convolutional networks on ImageNet and CIFAR10, achieving excellent accuracy on both training and test sets. They then took the same input images, but randomized the labels, and found that while their networks were now unable to generalize to the test set, they still memorized the training labels. They claimed these results contradict learning theory, although this claim is disputed (Kawaguchi et al., 2017; Dziugaite & Roy, 2017). Nonetheless, their results beg the question; if our models can assign arbitrary labels to the training set, why do they work so well in practice? Meanwhile Keskar et al. (2016) observed that if we hold the learning rate fixed and increase the batch size, the test accuracy usually falls. This striking result shows improving our estimate of the full-batch gradient can harm performance. Goyal et al. (2017) observed a linear scaling rule between batch size and learning rate in a deep ResNet, while Hoffer et al. (2017) proposed a square root rule on theoretical grounds.
+
+Many authors have suggested “broad minima” whose curvature is small may generalize better than “sharp minima” whose curvature is large (Chaudhari et al., 2016; Hochreiter & Schmidhuber, 1997). Indeed, Dziugaite & Roy (2017) argued the results of Zhang et al. (2016) can be understood using “nonvacuous” PAC-Bayes generalization bounds which penalize sharp minima, while Keskar et al. (2016) showed stochastic gradient descent (SGD) finds wider minima as the batch size is reduced. However Dinh et al. (2017) challenged this interpretation, by arguing that the curvature of a minimum can be arbitrarily increased by changing the model parameterization. In this work we show:
+
+• The results of Zhang et al. (2016) are not unique to deep learning; we observe the same phenomenon in a small “over-parameterized” linear model. We demonstrate that this phenomenon is straightforwardly understood by evaluating the Bayesian evidence in favor of each model, which penalizes sharp minima but is invariant to the model parameterization.
+
+• SGD integrates a stochastic differential equation whose “noise scale” $g \approx \epsilon N / B$ , where $\epsilon$ is the learning rate, $N$ training set size and $B$ batch size. Noise drives SGD away from sharp minima, and therefore there is an optimal batch size which maximizes the test set accuracy. This optimal batch size is proportional to the learning rate and training set size1.
+
+We describe Bayesian model comparison in section 2. In section 3 we replicate the observations of Zhang et al. (2016) in a linear model, and show they are explained by the Bayesian evidence. In section 4 we show there is an optimum batch size which maximizes the test set accuracy, and in section 5 we derive scaling rules between the optimum batch size, learning rate, training set size and momentum coefficient. Throughout this work, “generalization gap” refers to the gap in test accuracy between small and large batch SGD training, not the gap in accuracy between training and test sets.
+
+# 2 BAYESIAN MODEL COMPARISON
+
+Bayesian model comparison was first applied to neural networks in MacKay (1992). We provide a brief tutorial here, since the theory is central to the remainder of the paper. For simplicity we first consider a classification model $M$ with a single parameter $\omega$ , training inputs $x$ and training labels $y$ . We can infer a posterior probability distribution over the parameter by applying Bayes theorem,
+
+$$
+\begin{array} { r c l } { P ( \omega | y , x ; M ) } & { = } & { \frac { P ( y | \omega , x ; M ) P ( \omega ; M ) } { P ( y | x ; M ) } } \end{array}
+$$
+
+The likelihood, $\begin{array} { r c l c l } { P ( y | \omega , x ; M ) } & { = } & { \prod _ { i } P ( y _ { i } | \omega , x _ { i } ; M ) } & { = } & { e ^ { - H ( \omega ; M ) } } \end{array}$ , where $\begin{array} { r l } { H ( \omega ; M ) } & { { } = } \end{array}$ $\begin{array} { r } { - \sum _ { i } \ln \left( P ( y _ { i } | \omega , x _ { i } ; \dot { M } ) \right) } \end{array}$ denotes the cross-entropy of unique categorical labels. We typically use a Gaussian prior, $P ( \omega ; M ) = \sqrt { \lambda / 2 \pi } e ^ { - \lambda \omega ^ { 2 } / 2 }$ , and therefore the posterior probability density of the parameter given the training data, $P ( \omega | y , x ; M ) \propto \sqrt { \lambda / 2 \pi } e ^ { - C ( \omega ; M ) }$ , where $C ( \omega ; M ) =$ $H ( \omega ; M ) + \lambda \omega ^ { 2 } / 2$ denotes the L2 regularized cross entropy, or “cost function”, and $\lambda$ is the regularization coefficient. The value $\omega _ { 0 }$ which minimizes the cost function lies at the maximum of this posterior. To predict an unknown label $y _ { t }$ of a new input $x _ { t }$ , we should compute the integral,
+
+$$
+\begin{array} { r c l } { P ( y _ { t } | x _ { t } , y , x ; M ) } & { = } & { \displaystyle \int d \omega P ( y _ { t } | \omega , x _ { t } ; M ) P ( \omega | y , x ; M ) } \\ & { = } & { \displaystyle \frac { \int d \omega P ( y _ { t } | \omega , x _ { t } ; M ) e ^ { - C ( \omega ; M ) } } { \int d \omega e ^ { - C ( \omega ; M ) } } . } \end{array}
+$$
+
+However these integrals are dominated by the region near $\omega _ { 0 }$ , and since $P ( y _ { t } | \omega , x _ { t } ; M )$ is smooth we usually approximate $P ( y _ { t } | x _ { t } , x , y ; M ) \approx P ( y _ { t } | \omega _ { 0 } , x _ { t } ; M )$ . Having minimized $C ( \omega ; M )$ to find $\omega _ { 0 }$ , we now wish to compare two different models and select the best one. The probability ratio,
+
+$$
+\frac { P ( M _ { 1 } | y , x ) } { P ( M _ { 2 } | y , x ) } = \frac { P ( y | x ; M _ { 1 } ) } { P ( y | x ; M _ { 2 } ) } \frac { P ( M _ { 1 } ) } { P ( M _ { 2 } ) } .
+$$
+
+The second factor on the right is the prior ratio, which describes which model is most plausible. To avoid unnecessary subjectivity, we usually set this to 1. Meanwhile the first factor on the right is the evidence ratio, which controls how much the training data changes our prior beliefs. Germain et al. (2016) showed that maximizing the evidence (or “marginal likelihood”) minimizes a PAC-Bayes generalization bound. To compute it, we evaluate the normalizing constant of equation 1,
+
+$$
+\begin{array} { r c l } { { P ( y | x ; M ) } } & { { = } } & { { \displaystyle \int d \omega P ( y | \omega , x ; M ) P ( \omega ; M ) } } \\ { { } } & { { = } } & { { \displaystyle \sqrt { \frac { \lambda } { 2 \pi } } \int d \omega e ^ { - C ( \omega ; M ) } . } } \end{array}
+$$
+
+Notice that the evidence is computed by integrating out the parameters; and consequently it is invariant to the model parameterization. Since this integral is dominated by the region near the minimum $\omega _ { 0 }$ , we can estimate the evidence by Taylor expanding $C ( \omega ; M ) \approx \dot { C } ( \omega _ { 0 } ) + \dot { C } ^ { \prime \prime } ( \omega _ { 0 } ) ( \omega - \omega _ { 0 } ) ^ { 2 } / 2 .$ ,
+
+$$
+\begin{array} { r c l } { P ( y | x ; M ) } & { \approx } & { { { e } ^ { - C ( \omega _ { 0 } ) } } \displaystyle \sqrt { \frac { \lambda } { 2 \pi } } \int d \omega { e } ^ { - C ^ { \prime \prime } ( \omega _ { 0 } ) ( \omega - \omega _ { 0 } ) ^ { 2 } / 2 } } \\ & { = } & { \exp \bigg \{ - \bigg ( C ( \omega _ { 0 } ) + \frac { 1 } { 2 } \ln \left( C ^ { \prime \prime } ( \omega _ { 0 } ) / \lambda \right) \bigg ) \bigg \} . } \end{array}
+$$
+
+1Equivalently, there is an optimal learning rate proportional to the batch size and the training set size.
+
+Within this “Laplace” approximation, the evidence is controlled by the value of the cost function at the minimum, and by the logarithm of the ratio of the curvature about this minimum compared to the regularization constant. Thus far we have considered models of a single parameter; in realistic models with many parameters $P ( y | x ; M ) \approx \lambda ^ { \frac { p } { 2 } } e ^ { - C ( \omega _ { 0 } ) } / | \nabla \nabla C ( \omega ) | _ { \omega _ { 0 } } ^ { 1 / 2 }$ , where $| \nabla \nabla C ( \omega ) | _ { \omega _ { 0 } }$ is the determinant of the Hessian, and $p$ denotes the number of model parameters (Kass & Raftery, 1995). The determinant of the Hessian is simply the product of its eigenvalues, $( \prod _ { i = 1 } ^ { p } \lambda _ { i } )$ , and thus,
+
+$$
+P ( y | x ; M ) \approx \exp \Bigg \{ - \left( C ( \omega _ { 0 } ) + \frac { 1 } { 2 } \sum _ { i = 1 } ^ { p } \ln ( \lambda _ { i } / \lambda ) \right) \Bigg \} .
+$$
+
+The contribution $( \lambda ^ { \frac { p } { 2 } } / | \nabla \nabla C ( \omega ) | _ { \omega _ { 0 } } ^ { 1 / 2 } )$ is often called the “Occam factor”, because it enforces Occam’s razor; when two models describe the data equally well, the simpler model is usually better (Gull, 1988). Minima with low curvature are simple, because the parameters do not have to be finetuned to fit the data. Intuitively, the Occam factor describes the fraction of the prior parameter space consistent with the data. Since this fraction is always less than one, we propose to approximate equation 9 away from local minima by only performing the summation over eigenvalues $\lambda _ { i } \geq \lambda$ . The evidence can be reframed in the language of information theory, whereby Occam’s factor penalizes the amount of information the model must learn about the parameters to accurately model the training data (Hinton & Van Camp, 1993; Achille & Soatto, 2017; Shwartz-Ziv & Tishby, 2017).
+
+In this work, we will compare the evidence against a null model which assumes the labels are entirely random, assigning equal probability to each class. This unusual model has no parameters, and so the evidence is controlled by the likelihood alone, $P ( y | x ; N U L L ) = ( 1 / n ) ^ { N } \stackrel { - } { = } e ^ { - N \ln { ( n ) } }$ , where $n$ denotes the number of model classes and $N$ the number of training labels. Thus the evidence ratio,
+
+$$
+\frac { P ( y | x ; M ) } { P ( y | x ; N U L L ) } = e ^ { - E ( \omega _ { 0 } ) } ,
+$$
+
+Where $\begin{array} { r } { E ( \omega _ { 0 } ) = C ( \omega _ { 0 } ) + ( 1 / 2 ) \sum _ { i } \ln ( \lambda _ { i } / \lambda ) - N \ln ( n ) } \end{array}$ is the log evidence ratio in favor of the null model. Clearly, we should only assign any confidence to the predictions of our model if $E ( \omega _ { 0 } ) < 0$ .
+
+The evidence supports the intuition that broad minima generalize better than sharp minima, but unlike the curvature it does not depend on the model parameterization. Dinh et al. (2017) showed one can increase the Hessian eigenvalues by rescaling the parameters, but they must simultaneously rescale the regularization coefficients, otherwise the model changes. Since Occam’s factor arises from the log ratio, $\ln \left( \lambda _ { i } / \lambda \right)$ , these two effects cancel out2. It is difficult to evaluate the evidence for deep networks, as we cannot compute the Hessian of millions of parameters. Additionally, neural networks exhibit many equivalent minima, since we can permute the hidden units without changing the model. To compute the evidence we must carefully account for this “degeneracy”. We argue these issues are not a major limitation, since the intuition we build studying the evidence in simple cases will be sufficient to explain the results of both Zhang et al. (2016) and Keskar et al. (2016).
+
+# 3 BAYES THEOREM AND GENERALIZATION
+
+Zhang et al. (2016) showed that deep neural networks generalize well on training inputs with informative labels, and yet the same model can drastically overfit on the same input images when the labels are randomized; perfectly memorizing the training set. To demonstrate that these observations are not unique to deep networks, let’s consider a far simpler model; logistic regression. We form a small balanced training set comprising 800 images from MNIST, of which half have true label $\mathbf { \vec { \Delta } } ^ { 6 } 0 ^ { 9 }$ and half true label “1”. Our test set is also balanced, comprising 5000 MNIST images of zeros and 5000 MNIST images of ones. There are two tasks. In the first task, the labels of both the training and test sets are randomized. In the second task, the labels are informative, matching the true MNIST labels. Since the images contain 784 pixels, our model has just 784 weights and 1 bias.
+
+We show the accuracy of the model predictions on both the training and test sets in figure 1. When trained on the informative labels, the model generalizes well to the test set, so long as it is weakly regularized. However the model also perfectly memorizes the random labels, replicating the observations of Zhang et al. (2016) in deep networks. No significant improvement in model performance is observed as the regularization coefficient increases. For completeness, we also evaluate the mean margin between training examples and the decision boundary. For both random and informative labels, the margin drops significantly as we reduce the regularization coefficient. When weakly regularized, the mean margin is roughly $5 0 \%$ larger for informative labels than for random labels.
+
+![](images/b54a96e24409024f1dba39245529ce32aa588511ceab8602abf0ec1f279994eb.jpg)  
+Figure 1: Prediction accuracy and mean training set margin as a function of regularization coefficient, for a logistic regression trained on random (a) and informative (b) labels of the same inputs. The weakly regularized model generalizes well on informative labels but memorizes random labels.
+
+Now consider figure 2, where we plot the mean cross-entropy of the model predictions, evaluated on both training and test sets, as well as the Bayesian log evidence ratio defined in the previous section. Looking first at the random label experiment in figure 2a, while the cross-entropy on the training set vanishes when the model is weakly regularized, the cross-entropy on the test set explodes. Not only does the model make random predictions, but it is extremely confident in those predictions. As the regularization coefficient is increased the test set cross-entropy falls, settling at $\ln 2$ , the crossentropy of assigning equal probability to both classes. Now consider the Bayesian evidence, which we evaluate on the training set. The log evidence ratio is large and positive when the model is weakly regularized, indicating that the model is exponentially less plausible than assigning equal probabilities to each class. As the regularization parameter is increased, the log evidence ratio falls, but it is always positive, indicating that the model can never be expected to generalize well.
+
+Now consider figure 2b (informative labels). Once again, the training cross-entropy falls to zero when the model is weakly regularized, while the test cross-entropy is high. Even though the model makes accurate predictions, those predictions are overconfident. As the regularization coefficient increases, the test cross-entropy falls below $\ln 2$ , indicating that the model is successfully generalizing to the test set. Now consider the Bayesian evidence. The log evidence ratio is large and positive when the model is weakly regularized, but as the regularization coefficient increases, the log evidence ratio drops below zero, indicating that the model is exponentially more plausible than assigning equal probabilities to each class. As we further increase the regularization, the log evidence ratio rises to zero while the test cross-entropy rises to $\ln 2$ . Test cross-entropy and Bayesian evidence are strongly correlated, with minima at the same regularization strength.
+
+![](images/0f494f2c4f939d075fe13acb31a80bfe22b0322c78ac900bdee3b4d30d954879.jpg)  
+Figure 2: The cross-entropy and log evidence ratio, evaluated on random (a) or informative (b) labels. The evidence, evaluated on the training set, is strongly correlated with the test cross-entropy.
+
+Bayesian model comparison has explained our results in a logistic regression. Meanwhile, Krueger et al. (2017) showed the largest Hessian eigenvalue also increased when training on random labels in deep networks, implying the evidence is falling. We conclude that Bayesian model comparison is quantitatively consistent with the results of Zhang et al. (2016) in linear models where we can compute the evidence, and qualitatively consistent with their results in deep networks where we cannot. Dziugaite & Roy (2017) recently demonstrated the results of Zhang et al. (2016) can also be understood by minimising a PAC-Bayes generalization bound which penalizes sharp minima.
+
+# 4 BAYES THEOREM AND STOCHASTIC GRADIENT DESCENT
+
+We showed above that generalization is strongly correlated with the Bayesian evidence, a weighted combination of the depth of a minimum (the cost function) and its breadth (the Occam factor). Consequently Bayesians often add isotropic Gaussian noise to the gradient (Welling & Teh, 2011). In appendix A, we show this drives the parameters towards broad minima whose evidence is large. The noise introduced by small batch training is not isotropic, and its covariance matrix is a function of the parameter values, but empirically Keskar et al. (2016) found it has similar effects, driving the SGD away from sharp minima. This paper therefore proposes Bayesian principles also account for the “generalization gap”, whereby the test set accuracy often falls as the SGD batch size is increased (holding all other hyper-parameters constant). Since the gradient drives the SGD towards deep minima, while noise drives the SGD towards broad minima, we expect the test set performance to show a peak at an optimal batch size, which balances these competing contributions to the evidence.
+
+We were unable to observe a generalization gap in linear models (since linear models are convex there are no sharp minima to avoid). Instead we consider a shallow neural network with 800 hidden units and RELU hidden activations, trained on MNIST without regularization. We use SGD with a momentum parameter of 0.9. Unless otherwise stated, we use a constant learning rate of 1.0 which does not depend on the batch size or decay during training. Furthermore, we train on just 1000 images, selected at random from the MNIST training set. This enables us to compare small batch to full batch training. We emphasize that we are not trying to achieve optimal performance, but to study a simple model which shows a generalization gap between small and large batch training.
+
+In figure 3, we exhibit the evolution of the test accuracy and test cross-entropy during training. Our small batches are composed of 30 images, randomly sampled from the training set. Looking first at figure 3a, small batch training takes longer to converge, but after a thousand gradient updates a clear generalization gap in model accuracy emerges between small and large training batches. Now consider figure 3b. While the test cross-entropy for small batch training is lower at the end of training; the cross-entropy of both small and large training batches is increasing, indicative of over-fitting. Both models exhibit a minimum test cross-entropy, although after different numbers of gradient updates. Intriguingly, we show in appendix B that the generalization gap between small and large batch training shrinks significantly when we introduce L2 regularization.
+
+![](images/574a250cd29b8f09e8b5e4a578f9b9c70694cd5769423b16b172e086607be672.jpg)  
+Figure 3: The evolution during training of the test accuracy (a), and the test set cross-entropy (b). Full batches are composed of 1000 images, while small batches comprise 30 images.
+
+![](images/cb26d4d2e14c04d96237c32396fee654dc10720f0b6d94dc657d790de21c8f2c.jpg)  
+Figure 4: The test accuracy for a range of batch sizes, during training (a) and after 10000 steps (b).
+
+![](images/f3aa0fc16a2e26860bdf9e12234b09eee37b95e9b1f81d398ddde5651cd5ec96.jpg)  
+Figure 5: a) The test set accuracy as a function of batch size, for a range of learning rates $\epsilon$ . The performance peak shifts to the right as we increase $\epsilon$ , but the overall performance falls once $\epsilon \gtrsim 3$ . b) The best observed batch size is proportional to the learning rate across two orders of magnitude.
+
+From now on we focus on the test set accuracy (since this converges as the number of gradient updates increases). In figure 4a, we exhibit training curves for a range of batch sizes between 1 and 1000. We find that the model cannot train when the batch size $B \lessapprox \bar { 1 0 }$ . In figure 4b we plot the mean test set accuracy after 10000 training steps. A clear peak emerges, indicating that there is indeed an optimum batch size which maximizes the test accuracy, consistent with Bayesian intuition. The results of Keskar et al. (2016) focused on the decay in test accuracy above this optimum batch size.
+
+# 5 STOCHASTIC DIFFERENTIAL EQUATIONS AND THE SCALING RULES
+
+We showed above that the test accuracy peaks at an optimal batch size, if one holds the other SGD hyper-parameters constant. We argued that this peak arises from the tradeoff between depth and breadth in the Bayesian evidence. However it is not the batch size itself which controls this tradeoff, but the underlying scale of random fluctuations in the SGD dynamics. We now identify this SGD “noise scale”, and use it to derive three scaling rules which predict how the optimal batch size depends on the learning rate, training set size and momentum coefficient. A gradient update,
+
+$$
+\begin{array} { r c l } { \Delta \omega } & { = } & { - \displaystyle \frac { \epsilon } { N } \left( \frac { d C } { d \omega } + \left( \frac { d \hat { C } } { d \omega } - \frac { d C } { d \omega } \right) \right) , } \end{array}
+$$
+
+![](images/98b0bc094f9ff0d411a45850540a3de33d0a1a5554b328c700b05f4ca887c2a0.jpg)  
+Figure 6: a) The test accuracy as a function of batch size, for a range of training set sizes. To reduce noise, we average each curve over five experiments. The performance peak shift to the right as we increase the size of the training set. Unsurprisingly, the overall model performance also improves. b) The best observed batch size is proportional to the size of the training set once $N \gtrsim 2 0 0 0 \bar { 0 }$ .
+
+where  is the learning rate, N the training set size, dCdω $\begin{array} { r } { \frac { d C } { d \omega } = \sum _ { i = 1 } ^ { N } \frac { d C _ { i } } { d \omega } } \end{array}$ the true gradient, and $\begin{array} { r } { \frac { d \hat { C } } { d \omega } = } \end{array}$ NB PBi=1 dCidω the estimated gradient evaluated on a mini-batch. The expected gradient of a single example, $\begin{array} { r } { \left. \frac { d C _ { i } } { d \omega } \right. = \frac { 1 } { N } \frac { d C } { d \omega } } \end{array}$ , while $\begin{array} { r } { \left. \frac { d C _ { i } } { d \omega } \frac { d C _ { j } } { d \omega } \right. = \left( \frac { 1 } { N } \frac { d C } { d \omega } \right) ^ { 2 } + F ( \omega ) \delta _ { i j } . \mathrm { ~ } F ( \omega } \end{array}$ is a matrix describing the gradient covariances, which are a functheorem and model the gradient error $\begin{array} { r } { \alpha = ( \frac { d \hat { C } } { d \omega } - \frac { d C } { d \omega } ) } \end{array}$ nt par with alues. n rando approximation briefly in appendix C). It is easy to show that $\langle \alpha \rangle = 0$ $\begin{array} { r } { \langle \alpha ^ { 2 } \rangle = N ( \frac { N } { B } - 1 ) F ( \omega ) } \end{array}$ . Typically $N \gg B$ , such that $\langle \alpha ^ { 2 } \rangle \approx N ^ { 2 } F ( \omega ) / B$ . To continue, we interpret equation 11 as the discrete update of a stochastic differential equation (Li et al., 2017; Gardiner, 1985),
+
+$$
+\frac { d \omega } { d t } = - \frac { d C } { d \omega } + \eta ( t ) ,
+$$
+
+Where $t$ is a continuous variable, $\eta ( t )$ represents noise, $\langle \eta ( t ) \rangle = 0$ and $\langle \eta ( t ) \eta ( t ^ { \prime } ) \rangle = g F ( \omega ) \delta ( t { - } t ^ { \prime } )$ . The constant $g$ controls the scale of random fluctuations in the dynamics. To relate this differential equation to the SGD, we compute a gradient update $\begin{array} { r } { \Delta \omega = \int _ { 0 . } ^ { \epsilon / N } \frac { d \omega } { d t } d t = - \frac { \epsilon } { N } \frac { d C } { d \omega } + \int _ { 0 } ^ { \epsilon / N } \eta ( t ) d t } \end{array}$ Finally, to measure $\mathbf { g }$ , we equate the variance in this gradient update to the variance in equation 11,
+
+$$
+\begin{array} { r c l } { \displaystyle \left( \frac { \epsilon } { N } \right) ^ { 2 } \langle \alpha ^ { 2 } \rangle } & { = } & { \displaystyle \epsilon ^ { 2 } ( \frac { N } { B } - 1 ) F ( \omega ) / N } \\ { \displaystyle } & { = } & { \displaystyle \langle ( \int _ { 0 } ^ { \epsilon / N } d t \eta ( t ) ) ^ { 2 } \rangle = \int _ { 0 } ^ { \epsilon / N } d t \int _ { 0 } ^ { \epsilon / N } d t ^ { \prime } \langle \eta ( t ) \eta ( t ^ { \prime } ) \rangle = \epsilon g F ( \omega ) / N . } \end{array}
+$$
+
+Rearranging, the SGD noise scale $\begin{array} { r } { g = \epsilon ( \frac { N } { B } - 1 ) \approx \epsilon N / B } \end{array}$ . The noise scale falls when the batch size increases, consistent with our earlier observation of an optimal batch size $B _ { o p t }$ while holding the other hyper-parameters fixed. Notice that one would equivalently observe an optimal learning rate if one held the batch size constant. A similar analysis of the SGD was recently performed by Mandt et al. (2017), although their treatment only holds near local minima where the covariances $F ( \omega )$ are stationary. Our analysis holds throughout training, which is necessary since Keskar et al. (2016) found that the beneficial influence of noise was most pronounced at the start of training.
+
+When we vary the learning rate or the training set size, we should keep the noise scale fixed, which implies that $B _ { o p t } \propto \epsilon N$ . In figure 5a, we plot the test accuracy as a function of batch size after $( 1 0 0 0 0 / \epsilon )$ training steps, for a range of learning rates. Exactly as predicted, the peak moves to the right as $\epsilon$ increases. Additionally, the peak test accuracy achieved at a given learning rate does not begin to fall until $\epsilon \sim 3$ , indicating that there is no significant discretization error in integrating the stochastic differential equation below this point. Above this point, the discretization error begins to dominate and the peak test accuracy falls rapidly. In figure 5b, we plot the best observed batch size as a function of learning rate, observing a clear linear trend, $B _ { o p t } \propto \epsilon$ . The error bars indicate the distance from the best observed batch size to the next batch size sampled in our experiments.
+
+![](images/f65c1d948657f9b8bf2cf5faeff5b3749c8f5c3f8c2d509c3f217f125dee5dd5.jpg)  
+Figure 7: a) The test set accuracy as a function of batch size for a range of momentum coefficients. As expected, the peak moves to the right as the momentum coefficient increases. b) The best observed batch size for a range of momentum coefficients. The green curve exhibits the scaling rule.
+
+This scaling rule allows us to increase the learning rate with no loss in test accuracy and no increase in computational cost, simply by simultaneously increasing the batch size. We can then exploit increased parallelism across multiple GPUs, reducing model training times (Goyal et al., 2017). A similar scaling rule was independently proposed by Jastrzebski et al. (2017) and Chaudhari & Soatto (2017), although neither work identifies the existence of an optimal noise scale. A number of authors have proposed adjusting the batch size adaptively during training (Friedlander & Schmidt, 2012; Byrd et al., 2012; De et al., 2017), while Balles et al. (2016) proposed linearly coupling the learning rate and batch size within this framework. In Smith et al. (2017), we show empirically that decaying the learning rate during training and increasing the batch size during training are equivalent.
+
+In figure 6a we exhibit the test set accuracy as a function of batch size, for a range of training set sizes after 10000 steps $\mathbf { \epsilon } \cdot \mathbf { \epsilon } = 1$ everywhere). Once again, the peak shifts right as the training set size rises, although the generalization gap becomes less pronounced as the training set size increases. In figure 6b, we plot the best observed batch size as a function of training set size; observing another linear trend, $B _ { o p t } \propto N$ . This scaling rule could be applied to production models, progressively growing the batch size as new training data is collected. We expect production datasets to grow considerably over time, and consequently large batch training is likely to become increasingly common.
+
+Finally, in appendix $\mathrm { D }$ we extend our analysis to SGD with momentum, identifying the noise scale, $\begin{array} { r } { g \approx \frac { \epsilon N ^ { \mathrm { ~ \scriptsize ~ \cdots ~ } } } { B ( 1 - m ) } } \end{array}$ , where $m$ denotes the momentum coefficient. Notice that this reduces to the noise scale of conventional SGD as $m  0$ . When $m > 0$ , we obtain an additional scaling rule $B _ { o p t } \propto$ $1 / ( 1 - m )$ . This scaling rule predicts that the optimal batch size will increase when the momentum coefficient is increased. In figure 7a we plot the test set performance as a function of batch size after 10000 gradient updates ( $\mathbf { \epsilon } _ { \epsilon } = 1$ everywhere), for a range of momentum coefficients. In figure 7b, we plot the best observed batch size as a function of the momentum coefficient, and fit our results to the scaling rule above; obtaining remarkably good agreement. We propose a simple heuristic for tuning the batch size, learning rate and momentum coefficient in appendix E.
+
+# 6 CONCLUSIONS
+
+Just like deep neural networks, linear models which generalize well on informative labels can memorize random labels of the same inputs. These observations are explained by the Bayesian evidence, which is composed of the cost function and an “Occam factor”. The Occam factor penalizes sharp minima but it is invariant to changes in model parameterization. Mini-batch noise drives SGD away from sharp minima, and therefore there is an optimum batch size which maximizes the test accuracy. Interpreting SGD as the discretization of a stochastic differential equation, we predict this optimum batch size should scale linearly with both the learning rate and the training set size, $B _ { o p t } \propto \epsilon N$ . We derive an additional scaling rule, $B _ { o p t } \propto 1 / ( 1 - \stackrel { \textstyle - } { m } )$ , between the optimal batch size and the momentum coefficient. We verify these scaling rules empirically and discuss their implications.
+
+# ACKNOWLEDGMENTS
+
+We thank Pieter-Jan Kindermans, Prajit Ramachandran, Jascha Sohl-Dickstein, Jon Shlens, Kevin Murphy, Samy Bengio, Yasaman Bahri and Saeed Saremi for helpful comments on the manuscript.
+
+# REFERENCES
+
+Alessandro Achille and Stefano Soatto. On the emergence of invariance and disentangling in deep representations. arXiv preprint arXiv:1706.01350, 2017.
+
+Lukas Balles, Javier Romero, and Philipp Hennig. Coupling adaptive batch sizes with learning rates. arXiv preprint arXiv:1612.05086, 2016.
+
+Richard H Byrd, Gillian M Chin, Jorge Nocedal, and Yuchen Wu. Sample size selection in optimization methods for machine learning. Mathematical programming, 134(1):127–155, 2012.
+
+Pratik Chaudhari and Stefano Soatto. Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks. arXiv preprint arXiv:1710.11029, 2017.
+
+Pratik Chaudhari, Anna Choromanska, Stefano Soatto, and Yann LeCun. Entropy-SGD: Biasing gradient descent into wide valleys. arXiv preprint arXiv:1611.01838, 2016.
+
+Soham De, Abhay Yadav, David Jacobs, and Tom Goldstein. Automated inference with adaptive batches. In Artificial Intelligence and Statistics, pp. 1504–1513, 2017.
+
+Laurent Dinh, Razvan Pascanu, Samy Bengio, and Yoshua Bengio. Sharp minima can generalize for deep nets. arXiv preprint arXiv:1703.04933, 2017.
+
+Gintare Karolina Dziugaite and Daniel M Roy. Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. arXiv preprint arXiv:1703.11008, 2017.
+
+Michael P Friedlander and Mark Schmidt. Hybrid deterministic-stochastic methods for data fitting. SIAM Journal on Scientific Computing, 34(3):A1380–A1405, 2012.
+
+Crispin W Gardiner. Handbook of Stochastic Methods, volume 4. Springer Berlin, 1985.
+
+Pascal Germain, Francis Bach, Alexandre Lacoste, and Simon Lacoste-Julien. PAC-bayesian theory meets bayesian inference. In Advances in Neural Information Processing Systems, pp. 1884– 1892, 2016.
+
+Priya Goyal, Piotr Dollar, Ross Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, An- ´ drew Tulloch, Yangqing Jia, and Kaiming He. Accurate, large minibatch SGD: Training imagenet in 1 hour. arXiv preprint arXiv:1706.02677, 2017.
+
+Stephen F Gull. Bayesian inductive inference and maximum entropy. In Maximum-entropy and Bayesian methods in science and engineering, pp. 53–74. Springer, 1988.
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+Geoffrey E Hinton and Drew Van Camp. Keeping the neural networks simple by minimizing the description length of the weights. In Proceedings of the sixth annual conference on Computational learning theory, pp. 5–13. ACM, 1993.
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+Elad Hoffer, Itay Hubara, and Daniel Soudry. Train longer, generalize better: closing the generalization gap in large batch training of neural networks. arXiv preprint arXiv:1705.08741, 2017.
+
+Stanisław Jastrzebski, Zachary Kenton, Devansh Arpit, Nicolas Ballas, Asja Fischer, Yoshua Bengio, and Amos Storkey. Three factors influencing minima in SGD. arXiv preprint arXiv:1711.04623, 2017.
+
+Robert E Kass and Adrian E Raftery. Bayes factors. Journal of the american statistical association, 90(430):773–795, 1995.
+
+Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio. Generalization in deep learning. arXiv preprint arXiv:1710.05468, 2017.
+
+Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. arXiv preprint arXiv:1609.04836, 2016.
+
+David Krueger, Nicolas Ballas, Stanislaw Jastrzebski, Devansh Arpit, Maxinder S Kanwal, Tegan Maharaj, Emmanuel Bengio, Asja Fischer, and Aaron Courville. Deep nets don’t learn via memorization. ICLR Workshop, 2017.
+
+Qianxiao Li, Cheng Tai, and E Weinan. Stochastic modified equations and adaptive stochastic gradient algorithms. In International Conference on Machine Learning, pp. 2101–2110, 2017.
+
+David JC MacKay. A practical bayesian framework for backpropagation networks. Neural computation, 4(3):448–472, 1992.
+
+Stephan Mandt, Matthew D Hoffman, and David M Blei. Stochastic gradient descent as approximate bayesian inference. arXiv preprint arXiv:1704.04289, 2017.
+
+Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via information. arXiv preprint arXiv:1703.00810, 2017.
+
+Samuel L. Smith, Pieter-Jan Kindermans, and Quoc V. Le. Don’t decay the learning rate, increase the batch size. arXiv preprint arXiv:1711.00489, 2017.
+
+Max Welling and Yee W Teh. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th International Conference on Machine Learning (ICML-11), pp. 681–688, 2011.
+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.
+
+# A BAYESIAN POSTERIOR SAMPLING AND LANGEVIN DYNAMICS
+
+Instead of minimizing the cost function, Bayesian usually prefer to sample parameter values from the posterior (MacKay, 1992),
+
+$$
+P ( \omega | y , x ; M ) \propto e ^ { - C ( \omega ; M ) } ,
+$$
+
+where $C ( \omega ; M )$ is the regularized summed cost function, as shown in section 2 of the main text. It is well known that one can sample this posterior by simulating the overdamped Langevin equation (Gardiner, 1985), which is described by the stochastic differential equation,
+
+$$
+\frac { d \omega } { d t } = - \frac { d C } { d \omega } + \eta ( t ) ,
+$$
+
+where $t$ is a continuous variable, and $\eta ( t )$ describes Gaussian noise with mean $\langle \eta ( t ) \rangle = 0$ and variance $\langle \eta ( t ) \eta ( t ^ { \prime } ) \rangle = 2 T I \delta ( t - t ^ { \prime } )$ . The matrix $I$ denotes the identity, while $T$ is the “temperature”. Notice this Langevin equation is extremely similar to the stochastic differential equation of SGD, discussed in section 5 of the main text. Indeed, if the gradient covariances $F ( \omega )$ were stationary and proportional to the identity, then the SGD would integrate an overdamped Langevin equation with temperature proportional to the SGD noise scale $g$ . As $t \to \infty$ , the probability of sampling any particular parameter vector $\omega$ from the Langevin equation, $P ( \omega , t  \infty ) \propto e ^ { - C / \dot { T } }$ .
+
+We obtain posterior samples if $T = 1$ . In order to draw posterior samples in practice, we repeatedly integrate the Langevin equation (at temperature $T = 1$ ), over a finite step $t \to t + \epsilon / N$ ,
+
+$$
+\begin{array} { r c l } { \Delta \omega } & { = } & { - \displaystyle \frac { \epsilon } { N } \frac { d C } { d \omega } + \int _ { t } ^ { t + \frac { \epsilon } { N } } \eta ( t ) d t } \\ & { = } & { \displaystyle - \frac { \epsilon } { N } \frac { d C } { d \omega } + \alpha , } \end{array}
+$$
+
+where $\alpha$ denotes a Gaussian random variable with mean $\langle \alpha \rangle = 0$ and variance $\langle \alpha ^ { 2 } \rangle \ : = \ : 2 \epsilon I / N$ , which introduces isotropic noise to the gradient update as described in section 4 of the main text. Note that, since $C ( \omega ; M )$ denotes the summed cost function, we chose to scale our step size by the training set size $N$ . This also matches our treatment of SGD in section 5 of the main text. The larger the step size $\epsilon$ , the greater the discretization error, but if $\epsilon$ is sufficiently small and we iterate equation 17 sufficiently many times, we will obtain valid samples from the posterior.
+
+Since the probability of sampling any given parameter vector $\omega$ is proportional to the posterior, the probability of sampling a parameter vector belonging to any given local minimum is proportional to the integral of the posterior over the bowl of attraction $D$ which surrounds that minimum.
+
+$$
+P ( \omega \in D , t  \infty ) \propto \int _ { D } d \omega \ e ^ { - C ( \omega ; M ) } .
+$$
+
+Meanwhile we showed in section 2 of the main text that the evidence in favor of a model is proportional to the integral of the posterior over all parameter space.
+
+$$
+P ( y | x ; M ) \propto \int d \omega e ^ { - C ( \omega ; M ) } .
+$$
+
+As we discussed, this evidence is dominated by the contributions to the integral near local minima. In a convex model, there is only one such minimum; which allows us to accurately estimate the model evidence. Meanwhile, in non-convex models, there are many such minima, and so we can instead define the evidence as a sum over local evidences in favor of each minimum,
+
+$$
+P ( y | x ; M ) = \sum _ { i } P ( y | x , \omega \in D _ { i } ; M ) ,
+$$
+
+where we define the evidence in favor of a minimum as the integral over the local bowl of attraction,
+
+$$
+P ( y | x , \omega \in D ; M ) \propto \int _ { D } d \omega \ e ^ { - C ( \omega ; M ) } .
+$$
+
+Since the combined bowls of attraction of all the minima perfectly tile the entire parameter space, equations 19 and 20 are equivalent. Meanwhile, equating equations 18 and 21 we find that, when one performs Bayesian posterior sampling, the probability of sampling the parameter vector from a local minimum is proportional to the evidence in favor of that minimum. This demonstrates that bayesian posterior samples are biased in favor of local minima whose evidence is large, which explains why a single posterior sample $\omega _ { p }$ often achieves lower test error than the cost function minimum $\omega _ { 0 }$ .
+
+# B THE EFFECT OF REGULARIZATION ON THE GENERALIZATION GAP
+
+In the experiments of section 4 of the main text, the L2 regularization coefficient $\lambda = 0$ . In figure 8, we plot the evolution of the training curves when $\lambda = 0 . 1$ , for both small batch and full batch training. Excluding the regularization parameter, these experiments are identical to figure 3. To our surprise, regularized full batch training took longer to converge than small batch training. In another surprise, regularization significantly reduced the size of the generalization gap. While large batch regularized training achieves slightly lower test set accuracy than unregularized small batch training, it also achieves lower test cross-entropy. The test cross-entropy of our regularized models does not degrade after many gradient updates, removing the need for early stopping.
+
+![](images/9b725df1b2851f06eebf187dcc70cd8ce8c558a8b2aed25f92d9127ad2a42bfe.jpg)  
+Figure 8: The mean test accuracy (a) and the mean test cross-entropy (b) of a regularized model during training. While full batch training takes longer to converge, it achieves similar performance at long times. The noise inherent in small batch training causes the performance to fluctuate.
+
+![](images/0e9f4bc9e0527e3032e6c0929c574be057d50f2e1d13b90d5cc6350b4fcdf353.jpg)  
+Figure 9: The gradient distribution of a randomly selected parameter in the softmax layer, when measured over a single training example (a), and when averaged over mini-batches of 30 images (b).
+
+# C THE GAUSSIAN APPROXIMATION TO THE MINI-BATCH ERROR
+
+In section 5 of the main text, we approximated the difference between the full batch gradient and the mini-batch gradient estimate, $\begin{array} { r } { \alpha = ( \frac { d \hat { C } } { d \omega } - \frac { d C } { d \omega } ) } \end{array}$ , by a Gaussian random variable. This enabled us to derive the scaling rules, which we verified empirically. We motivated this assumption by reference to the central limit theorem, which states that the gradient error will tend towards Gaussian noise as $\{ N \to \infty , B \to \infty , B \ll N \}$ , so long as the distribution of gradients over individual training examples does not have heavy tails. In practice neither $\mathbf { N }$ nor B is infinite, and the gradient distribution may be heavy tailed, especially when gradients are sparse. Nonetheless the central limit theorem tends to be surprisingly robust in practice, and is consequently widely used.
+
+It is beyond the scope of this work to perform a thorough study of the gradient noise distribution in deep networks. However as a brief proof of principle, we present the distribution of the gradient immediately after random initialization in figure 9, for the shallow neural network discussed in sections 4 and 5 of the main text. In figure 9a, we present the distribution over the individual training examples, of the gradient of a single matrix element in the softmax output layer, chosen randomly. The distribution is double peaked and clearly not Gaussian. However in figure 7b, we plot the distribution of the gradient of the same matrix element, when averaged over randomly sampled mini-batches of 30 images (without replacement). A single peak emerges, and while the distribution is still slightly skewed, it is clearly already approaching the Gaussian limit. We conclude that the Gaussian approximation is likely to be reasonable for commonly used mini-batch sizes.
+
+# D DERIVING THE SCALING RULES FOR SGD WITH MOMENTUM
+
+Momentum simulates a generalized Langevin equation (with structured fluctuations),
+
+$$
+\frac { d ^ { 2 } \omega } { d t ^ { 2 } } = - \lambda \frac { d \omega } { d t } - \frac { d C } { d \omega } + \eta ( t ) .
+$$
+
+$\lambda$ is the “damping coefficient” and $\eta ( t )$ describes Gaussian noise, whose statistics $\langle \eta ( t ) \rangle = 0$ and $\langle \eta ( t ) \eta ( t ^ { \prime } ) \rangle = \bar { g } \lambda \bar { F } ( w ) \delta ( t - t ^ { \prime } )$ . As before, the coefficient $g$ describes the scale of random fluctuations in the dynamics, and $F ( \omega )$ describes the gradient covariances between parameters. We include a factor of $\lambda$ in the noise variance to satisfy the fluctuation-dissipation theorem, which states that we can vary the damping coefficient without changing the probability of sampling any particular configuration of parameters in the limit $t \to \infty$ , if we proportionally increase the noise variance.
+
+To relate this Langevin equation to the usual momentum equations, we first re-express it as two coupled first order differential equations,
+
+$$
+\begin{array} { r c l } { \displaystyle \frac { d p } { d t } } & { = } & { \displaystyle - \lambda p - \frac { d C } { d \omega } + \eta ( t ) , } \\ { \displaystyle \frac { d \omega } { d t } } & { = } & { p . } \end{array}
+$$
+
+Integrating over a single step $\Delta t / N$ ,
+
+$$
+\begin{array} { r c l } { { \Delta p } } & { { = } } & { { \displaystyle - ( \lambda \Delta t / N ) p - \frac { \Delta t } { N } \frac { d C } { d \omega } + \eta , } } \\ { { \Delta \omega } } & { { = } } & { { p \Delta t / N . } } \end{array}
+$$
+
+Where now $\langle \eta \rangle = 0$ and $\langle \eta ^ { 2 } \rangle = g \Delta t \lambda F ( w ) / N$ . We define the accumulation $A = p / \Delta t$ ,
+
+$$
+\begin{array} { r c l } { \Delta A } & { = } & { - ( \lambda \Delta t / N ) A - \displaystyle \frac { 1 } { N } \frac { d C } { d \omega } + \frac { \eta } { \Delta t } , } \\ { \Delta \omega } & { = } & { ( \Delta t ) ^ { 2 } A / N . } \end{array}
+$$
+
+These equations can be compared to the TensorFlow update equations for momentum,
+
+$$
+\begin{array} { r c l } { { \Delta A } } & { { = } } & { { ( m - 1 ) A - \displaystyle \frac { 1 } { N } \left( \displaystyle \frac { d C } { d \omega } + \alpha \right) , } } \\ { { \Delta \omega } } & { { = } } & { { \epsilon A . } } \end{array}
+$$
+
+Where $\begin{array} { r } { \alpha = \left( \frac { d \hat { C } } { d \omega } - \frac { d C } { d \omega } \right) } \end{array}$ denotes the error in the gradient update. As discussed in the main text, we can approximate this error as Gaussian noise with statistics $\langle \alpha \rangle = 0$ and $\langle \alpha ^ { 2 } \rangle \approx N ^ { 2 } F ( \omega ) / B$ . Equations 27 and 28 match equations 29 and 30 if the step size $\dot { \epsilon } = ( \Delta t ) ^ { 2 } / N$ , and the momentum parameter $m = ( 1 - \lambda \Delta t / N )$ . Finally we equate the noise by setting $\langle \alpha ^ { 2 } \rangle / N ^ { 2 } = \langle \eta ^ { 2 } \rangle / ( \Delta t ) ^ { 2 }$ , and solve for the noise scale $g$ to obtain,
+
+$$
+\begin{array} { l l l } { { g } } & { { \approx } } & { { \displaystyle \frac { \Delta t N } { B \lambda } = \frac { ( \Delta t ) ^ { 2 } } { B ( 1 - m ) } } } \\ { { } } & { { = } } & { { \displaystyle \frac { \epsilon N } { B ( 1 - m ) } . } } \end{array}
+$$
+
+As observed in the main text, if we wish to keep the scale of random fluctuations constant, then we should scale the batch size $B \propto \epsilon N$ . We also predict an additional scaling relation between the batch size and the momentum parameter, $B \propto \mathsf { \bar { 1 } } / ( 1 - m )$ . Note that one can also interpret $\epsilon _ { e f f } = \epsilon / ( 1 - m )$ as the “effective learning rate”.
+
+# E HOW TO ACHIEVE LARGE BATCH TRAINING
+
+Here we propose a simple heuristic for tuning the batch size, learning rate and momentum parameter; in order to maximize both test accuracy and batch size (enabling parallel training across many machines). Note that this is only worthwhile if one expects to retrain a model many times.
+
+1. Set the learning rate to 0.1 and the momentum coefficient to 0.9. Run experiments at a range of batch sizes on a logarithmic scale, and identify the optimal batch size which maximizes the validation set accuracy. If training is not stable, reduce the learning rate, and repeat. 2. Repeatedly increase the batch size by a factor of 3, while scaling the learning rate $\epsilon \propto B$ , until the validation set accuracy starts to fall. Then repeatedly increase the batch size by a factor of 3, while scaling the momentum coefficient $( 1 - \dot { m } ) \propto 1 / B$ , until either the validation set accuracy falls or the batch size reaches the limits of your hardware. 3. Having identified the final learning rate and momentum parameter, retune the batch size on a linear scale in the local neighborhood of the current batch size.
+
+We believe that this simple procedure will increase the test accuracy, reduce the cost of tuning hyperparameters, and significantly reduce the final number of gradient updates required to train a model.

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+    {
+        "type": "text",
+        "text": "A BAYESIAN PERSPECTIVE ON GENERALIZATION AND STOCHASTIC GRADIENT DESCENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Samuel L. Smith∗ & Quoc V. Le Google Brain {slsmith, qvl}@google.com ",
+        "bbox": [
+            183,
+            170,
+            428,
+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider two questions at the heart of machine learning; how can we predict if a minimum will generalize to the test set, and why does stochastic gradient descent find minima that generalize well? Our work responds to Zhang et al. (2016), who showed deep neural networks can easily memorize randomly labeled training data, despite generalizing well on real labels of the same inputs. We show that the same phenomenon occurs in small linear models. These observations are explained by the Bayesian evidence, which penalizes sharp minima but is invariant to model parameterization. We also demonstrate that, when one holds the learning rate fixed, there is an optimum batch size which maximizes the test set accuracy. We propose that the noise introduced by small mini-batches drives the parameters towards minima whose evidence is large. Interpreting stochastic gradient descent as a stochastic differential equation, we identify the “noise scale” $\\begin{array} { r } { \\overline { { g } } = \\epsilon ( \\frac { N } { B } - 1 ) \\approx } \\end{array}$ $\\epsilon N / B$ , where $\\epsilon$ is the learning rate, $N$ the training set size and $B$ the batch size. Consequently the optimum batch size is proportional to both the learning rate and the size of the training set, $B _ { o p t } \\propto \\epsilon N$ . We verify these predictions empirically. ",
+        "bbox": [
+            233,
+            282,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            521,
+            334,
+            537
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper shows Bayesian principles can explain many recent observations in the deep learning literature, while also discovering practical new insights. Zhang et al. (2016) trained deep convolutional networks on ImageNet and CIFAR10, achieving excellent accuracy on both training and test sets. They then took the same input images, but randomized the labels, and found that while their networks were now unable to generalize to the test set, they still memorized the training labels. They claimed these results contradict learning theory, although this claim is disputed (Kawaguchi et al., 2017; Dziugaite & Roy, 2017). Nonetheless, their results beg the question; if our models can assign arbitrary labels to the training set, why do they work so well in practice? Meanwhile Keskar et al. (2016) observed that if we hold the learning rate fixed and increase the batch size, the test accuracy usually falls. This striking result shows improving our estimate of the full-batch gradient can harm performance. Goyal et al. (2017) observed a linear scaling rule between batch size and learning rate in a deep ResNet, while Hoffer et al. (2017) proposed a square root rule on theoretical grounds. ",
+        "bbox": [
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+            554,
+            825,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many authors have suggested “broad minima” whose curvature is small may generalize better than “sharp minima” whose curvature is large (Chaudhari et al., 2016; Hochreiter & Schmidhuber, 1997). Indeed, Dziugaite & Roy (2017) argued the results of Zhang et al. (2016) can be understood using “nonvacuous” PAC-Bayes generalization bounds which penalize sharp minima, while Keskar et al. (2016) showed stochastic gradient descent (SGD) finds wider minima as the batch size is reduced. However Dinh et al. (2017) challenged this interpretation, by arguing that the curvature of a minimum can be arbitrarily increased by changing the model parameterization. In this work we show: ",
+        "bbox": [
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+            728,
+            825,
+            825
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• The results of Zhang et al. (2016) are not unique to deep learning; we observe the same phenomenon in a small “over-parameterized” linear model. We demonstrate that this phenomenon is straightforwardly understood by evaluating the Bayesian evidence in favor of each model, which penalizes sharp minima but is invariant to the model parameterization. ",
+        "bbox": [
+            217,
+            840,
+            823,
+            896
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• SGD integrates a stochastic differential equation whose “noise scale” $g \\approx \\epsilon N / B$ , where $\\epsilon$ is the learning rate, $N$ training set size and $B$ batch size. Noise drives SGD away from sharp minima, and therefore there is an optimal batch size which maximizes the test set accuracy. This optimal batch size is proportional to the learning rate and training set size1. ",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We describe Bayesian model comparison in section 2. In section 3 we replicate the observations of Zhang et al. (2016) in a linear model, and show they are explained by the Bayesian evidence. In section 4 we show there is an optimum batch size which maximizes the test set accuracy, and in section 5 we derive scaling rules between the optimum batch size, learning rate, training set size and momentum coefficient. Throughout this work, “generalization gap” refers to the gap in test accuracy between small and large batch SGD training, not the gap in accuracy between training and test sets. ",
+        "bbox": [
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+            169,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BAYESIAN MODEL COMPARISON ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            273,
+            473,
+            289
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Bayesian model comparison was first applied to neural networks in MacKay (1992). We provide a brief tutorial here, since the theory is central to the remainder of the paper. For simplicity we first consider a classification model $M$ with a single parameter $\\omega$ , training inputs $x$ and training labels $y$ . We can infer a posterior probability distribution over the parameter by applying Bayes theorem, ",
+        "bbox": [
+            173,
+            303,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/638dd4ddf5315d75a1652360e2d31405a240f2094aae9efc929ab7fea5338e6a.jpg",
+        "text": "$$\n\\begin{array} { r c l } { P ( \\omega | y , x ; M ) } & { = } & { \\frac { P ( y | \\omega , x ; M ) P ( \\omega ; M ) } { P ( y | x ; M ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The likelihood, $\\begin{array} { r c l c l } { P ( y | \\omega , x ; M ) } & { = } & { \\prod _ { i } P ( y _ { i } | \\omega , x _ { i } ; M ) } & { = } & { e ^ { - H ( \\omega ; M ) } } \\end{array}$ , where $\\begin{array} { r l } { H ( \\omega ; M ) } & { { } = } \\end{array}$ $\\begin{array} { r } { - \\sum _ { i } \\ln \\left( P ( y _ { i } | \\omega , x _ { i } ; \\dot { M } ) \\right) } \\end{array}$ denotes the cross-entropy of unique categorical labels. We typically use a Gaussian prior, $P ( \\omega ; M ) = \\sqrt { \\lambda / 2 \\pi } e ^ { - \\lambda \\omega ^ { 2 } / 2 }$ , and therefore the posterior probability density of the parameter given the training data, $P ( \\omega | y , x ; M ) \\propto \\sqrt { \\lambda / 2 \\pi } e ^ { - C ( \\omega ; M ) }$ , where $C ( \\omega ; M ) =$ $H ( \\omega ; M ) + \\lambda \\omega ^ { 2 } / 2$ denotes the L2 regularized cross entropy, or “cost function”, and $\\lambda$ is the regularization coefficient. The value $\\omega _ { 0 }$ which minimizes the cost function lies at the maximum of this posterior. To predict an unknown label $y _ { t }$ of a new input $x _ { t }$ , we should compute the integral, ",
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+        "type": "equation",
+        "img_path": "images/5a66ef55f712d0bbc7f7a762975b13c55f437b17b7e6894f815d285a9f163ce5.jpg",
+        "text": "$$\n\\begin{array} { r c l } { P ( y _ { t } | x _ { t } , y , x ; M ) } & { = } & { \\displaystyle \\int d \\omega P ( y _ { t } | \\omega , x _ { t } ; M ) P ( \\omega | y , x ; M ) } \\\\ & { = } & { \\displaystyle \\frac { \\int d \\omega P ( y _ { t } | \\omega , x _ { t } ; M ) e ^ { - C ( \\omega ; M ) } } { \\int d \\omega e ^ { - C ( \\omega ; M ) } } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "However these integrals are dominated by the region near $\\omega _ { 0 }$ , and since $P ( y _ { t } | \\omega , x _ { t } ; M )$ is smooth we usually approximate $P ( y _ { t } | x _ { t } , x , y ; M ) \\approx P ( y _ { t } | \\omega _ { 0 } , x _ { t } ; M )$ . Having minimized $C ( \\omega ; M )$ to find $\\omega _ { 0 }$ , we now wish to compare two different models and select the best one. The probability ratio, ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/dcefd7747160699b3d2c79f64482749cf659571784bd1b9a54cfb67fdc34e033.jpg",
+        "text": "$$\n\\frac { P ( M _ { 1 } | y , x ) } { P ( M _ { 2 } | y , x ) } = \\frac { P ( y | x ; M _ { 1 } ) } { P ( y | x ; M _ { 2 } ) } \\frac { P ( M _ { 1 } ) } { P ( M _ { 2 } ) } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The second factor on the right is the prior ratio, which describes which model is most plausible. To avoid unnecessary subjectivity, we usually set this to 1. Meanwhile the first factor on the right is the evidence ratio, which controls how much the training data changes our prior beliefs. Germain et al. (2016) showed that maximizing the evidence (or “marginal likelihood”) minimizes a PAC-Bayes generalization bound. To compute it, we evaluate the normalizing constant of equation 1, ",
+        "bbox": [
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+        "page_idx": 1
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+        "type": "equation",
+        "img_path": "images/7273c681ce2981db1167b6c0ac81b39d37907cba1378976e53e9f1d2f50ba37e.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { P ( y | x ; M ) } } & { { = } } & { { \\displaystyle \\int d \\omega P ( y | \\omega , x ; M ) P ( \\omega ; M ) } } \\\\ { { } } & { { = } } & { { \\displaystyle \\sqrt { \\frac { \\lambda } { 2 \\pi } } \\int d \\omega e ^ { - C ( \\omega ; M ) } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Notice that the evidence is computed by integrating out the parameters; and consequently it is invariant to the model parameterization. Since this integral is dominated by the region near the minimum $\\omega _ { 0 }$ , we can estimate the evidence by Taylor expanding $C ( \\omega ; M ) \\approx \\dot { C } ( \\omega _ { 0 } ) + \\dot { C } ^ { \\prime \\prime } ( \\omega _ { 0 } ) ( \\omega - \\omega _ { 0 } ) ^ { 2 } / 2 .$ , ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/4d35e3757beb121e0bd4a872e8801447132e1c4aea77012af4697d8b49190471.jpg",
+        "text": "$$\n\\begin{array} { r c l } { P ( y | x ; M ) } & { \\approx } & { { { e } ^ { - C ( \\omega _ { 0 } ) } } \\displaystyle \\sqrt { \\frac { \\lambda } { 2 \\pi } } \\int d \\omega { e } ^ { - C ^ { \\prime \\prime } ( \\omega _ { 0 } ) ( \\omega - \\omega _ { 0 } ) ^ { 2 } / 2 } } \\\\ & { = } & { \\exp \\bigg \\{ - \\bigg ( C ( \\omega _ { 0 } ) + \\frac { 1 } { 2 } \\ln \\left( C ^ { \\prime \\prime } ( \\omega _ { 0 } ) / \\lambda \\right) \\bigg ) \\bigg \\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1Equivalently, there is an optimal learning rate proportional to the batch size and the training set size. ",
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+        "type": "text",
+        "text": "Within this “Laplace” approximation, the evidence is controlled by the value of the cost function at the minimum, and by the logarithm of the ratio of the curvature about this minimum compared to the regularization constant. Thus far we have considered models of a single parameter; in realistic models with many parameters $P ( y | x ; M ) \\approx \\lambda ^ { \\frac { p } { 2 } } e ^ { - C ( \\omega _ { 0 } ) } / | \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } } ^ { 1 / 2 }$ , where $| \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } }$ is the determinant of the Hessian, and $p$ denotes the number of model parameters (Kass & Raftery, 1995). The determinant of the Hessian is simply the product of its eigenvalues, $( \\prod _ { i = 1 } ^ { p } \\lambda _ { i } )$ , and thus, ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/4078e8618d773291ae1f3438d48fd4149d358d3f863d89216c7f967746d94848.jpg",
+        "text": "$$\nP ( y | x ; M ) \\approx \\exp \\Bigg \\{ - \\left( C ( \\omega _ { 0 } ) + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { p } \\ln ( \\lambda _ { i } / \\lambda ) \\right) \\Bigg \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The contribution $( \\lambda ^ { \\frac { p } { 2 } } / | \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } } ^ { 1 / 2 } )$ is often called the “Occam factor”, because it enforces Occam’s razor; when two models describe the data equally well, the simpler model is usually better (Gull, 1988). Minima with low curvature are simple, because the parameters do not have to be finetuned to fit the data. Intuitively, the Occam factor describes the fraction of the prior parameter space consistent with the data. Since this fraction is always less than one, we propose to approximate equation 9 away from local minima by only performing the summation over eigenvalues $\\lambda _ { i } \\geq \\lambda$ . The evidence can be reframed in the language of information theory, whereby Occam’s factor penalizes the amount of information the model must learn about the parameters to accurately model the training data (Hinton & Van Camp, 1993; Achille & Soatto, 2017; Shwartz-Ziv & Tishby, 2017). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In this work, we will compare the evidence against a null model which assumes the labels are entirely random, assigning equal probability to each class. This unusual model has no parameters, and so the evidence is controlled by the likelihood alone, $P ( y | x ; N U L L ) = ( 1 / n ) ^ { N } \\stackrel { - } { = } e ^ { - N \\ln { ( n ) } }$ , where $n$ denotes the number of model classes and $N$ the number of training labels. Thus the evidence ratio, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/308e11f4919465f708f251adcb3731dc2ceb7ea6fcb80afc3aa7e9027ac7d11e.jpg",
+        "text": "$$\n\\frac { P ( y | x ; M ) } { P ( y | x ; N U L L ) } = e ^ { - E ( \\omega _ { 0 } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Where $\\begin{array} { r } { E ( \\omega _ { 0 } ) = C ( \\omega _ { 0 } ) + ( 1 / 2 ) \\sum _ { i } \\ln ( \\lambda _ { i } / \\lambda ) - N \\ln ( n ) } \\end{array}$ is the log evidence ratio in favor of the null model. Clearly, we should only assign any confidence to the predictions of our model if $E ( \\omega _ { 0 } ) < 0$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The evidence supports the intuition that broad minima generalize better than sharp minima, but unlike the curvature it does not depend on the model parameterization. Dinh et al. (2017) showed one can increase the Hessian eigenvalues by rescaling the parameters, but they must simultaneously rescale the regularization coefficients, otherwise the model changes. Since Occam’s factor arises from the log ratio, $\\ln \\left( \\lambda _ { i } / \\lambda \\right)$ , these two effects cancel out2. It is difficult to evaluate the evidence for deep networks, as we cannot compute the Hessian of millions of parameters. Additionally, neural networks exhibit many equivalent minima, since we can permute the hidden units without changing the model. To compute the evidence we must carefully account for this “degeneracy”. We argue these issues are not a major limitation, since the intuition we build studying the evidence in simple cases will be sufficient to explain the results of both Zhang et al. (2016) and Keskar et al. (2016). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3 BAYES THEOREM AND GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Zhang et al. (2016) showed that deep neural networks generalize well on training inputs with informative labels, and yet the same model can drastically overfit on the same input images when the labels are randomized; perfectly memorizing the training set. To demonstrate that these observations are not unique to deep networks, let’s consider a far simpler model; logistic regression. We form a small balanced training set comprising 800 images from MNIST, of which half have true label $\\mathbf { \\vec { \\Delta } } ^ { 6 } 0 ^ { 9 }$ and half true label “1”. Our test set is also balanced, comprising 5000 MNIST images of zeros and 5000 MNIST images of ones. There are two tasks. In the first task, the labels of both the training and test sets are randomized. In the second task, the labels are informative, matching the true MNIST labels. Since the images contain 784 pixels, our model has just 784 weights and 1 bias. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show the accuracy of the model predictions on both the training and test sets in figure 1. When trained on the informative labels, the model generalizes well to the test set, so long as it is weakly regularized. However the model also perfectly memorizes the random labels, replicating the observations of Zhang et al. (2016) in deep networks. No significant improvement in model performance is observed as the regularization coefficient increases. For completeness, we also evaluate the mean margin between training examples and the decision boundary. For both random and informative labels, the margin drops significantly as we reduce the regularization coefficient. When weakly regularized, the mean margin is roughly $5 0 \\%$ larger for informative labels than for random labels. ",
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+    {
+        "type": "image",
+        "img_path": "images/b54a96e24409024f1dba39245529ce32aa588511ceab8602abf0ec1f279994eb.jpg",
+        "image_caption": [
+            "Figure 1: Prediction accuracy and mean training set margin as a function of regularization coefficient, for a logistic regression trained on random (a) and informative (b) labels of the same inputs. The weakly regularized model generalizes well on informative labels but memorizes random labels. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Now consider figure 2, where we plot the mean cross-entropy of the model predictions, evaluated on both training and test sets, as well as the Bayesian log evidence ratio defined in the previous section. Looking first at the random label experiment in figure 2a, while the cross-entropy on the training set vanishes when the model is weakly regularized, the cross-entropy on the test set explodes. Not only does the model make random predictions, but it is extremely confident in those predictions. As the regularization coefficient is increased the test set cross-entropy falls, settling at $\\ln 2$ , the crossentropy of assigning equal probability to both classes. Now consider the Bayesian evidence, which we evaluate on the training set. The log evidence ratio is large and positive when the model is weakly regularized, indicating that the model is exponentially less plausible than assigning equal probabilities to each class. As the regularization parameter is increased, the log evidence ratio falls, but it is always positive, indicating that the model can never be expected to generalize well. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Now consider figure 2b (informative labels). Once again, the training cross-entropy falls to zero when the model is weakly regularized, while the test cross-entropy is high. Even though the model makes accurate predictions, those predictions are overconfident. As the regularization coefficient increases, the test cross-entropy falls below $\\ln 2$ , indicating that the model is successfully generalizing to the test set. Now consider the Bayesian evidence. The log evidence ratio is large and positive when the model is weakly regularized, but as the regularization coefficient increases, the log evidence ratio drops below zero, indicating that the model is exponentially more plausible than assigning equal probabilities to each class. As we further increase the regularization, the log evidence ratio rises to zero while the test cross-entropy rises to $\\ln 2$ . Test cross-entropy and Bayesian evidence are strongly correlated, with minima at the same regularization strength. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "image",
+        "img_path": "images/0f494f2c4f939d075fe13acb31a80bfe22b0322c78ac900bdee3b4d30d954879.jpg",
+        "image_caption": [
+            "Figure 2: The cross-entropy and log evidence ratio, evaluated on random (a) or informative (b) labels. The evidence, evaluated on the training set, is strongly correlated with the test cross-entropy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Bayesian model comparison has explained our results in a logistic regression. Meanwhile, Krueger et al. (2017) showed the largest Hessian eigenvalue also increased when training on random labels in deep networks, implying the evidence is falling. We conclude that Bayesian model comparison is quantitatively consistent with the results of Zhang et al. (2016) in linear models where we can compute the evidence, and qualitatively consistent with their results in deep networks where we cannot. Dziugaite & Roy (2017) recently demonstrated the results of Zhang et al. (2016) can also be understood by minimising a PAC-Bayes generalization bound which penalizes sharp minima. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4 BAYES THEOREM AND STOCHASTIC GRADIENT DESCENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We showed above that generalization is strongly correlated with the Bayesian evidence, a weighted combination of the depth of a minimum (the cost function) and its breadth (the Occam factor). Consequently Bayesians often add isotropic Gaussian noise to the gradient (Welling & Teh, 2011). In appendix A, we show this drives the parameters towards broad minima whose evidence is large. The noise introduced by small batch training is not isotropic, and its covariance matrix is a function of the parameter values, but empirically Keskar et al. (2016) found it has similar effects, driving the SGD away from sharp minima. This paper therefore proposes Bayesian principles also account for the “generalization gap”, whereby the test set accuracy often falls as the SGD batch size is increased (holding all other hyper-parameters constant). Since the gradient drives the SGD towards deep minima, while noise drives the SGD towards broad minima, we expect the test set performance to show a peak at an optimal batch size, which balances these competing contributions to the evidence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We were unable to observe a generalization gap in linear models (since linear models are convex there are no sharp minima to avoid). Instead we consider a shallow neural network with 800 hidden units and RELU hidden activations, trained on MNIST without regularization. We use SGD with a momentum parameter of 0.9. Unless otherwise stated, we use a constant learning rate of 1.0 which does not depend on the batch size or decay during training. Furthermore, we train on just 1000 images, selected at random from the MNIST training set. This enables us to compare small batch to full batch training. We emphasize that we are not trying to achieve optimal performance, but to study a simple model which shows a generalization gap between small and large batch training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In figure 3, we exhibit the evolution of the test accuracy and test cross-entropy during training. Our small batches are composed of 30 images, randomly sampled from the training set. Looking first at figure 3a, small batch training takes longer to converge, but after a thousand gradient updates a clear generalization gap in model accuracy emerges between small and large training batches. Now consider figure 3b. While the test cross-entropy for small batch training is lower at the end of training; the cross-entropy of both small and large training batches is increasing, indicative of over-fitting. Both models exhibit a minimum test cross-entropy, although after different numbers of gradient updates. Intriguingly, we show in appendix B that the generalization gap between small and large batch training shrinks significantly when we introduce L2 regularization. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/574a250cd29b8f09e8b5e4a578f9b9c70694cd5769423b16b172e086607be672.jpg",
+        "image_caption": [
+            "Figure 3: The evolution during training of the test accuracy (a), and the test set cross-entropy (b). Full batches are composed of 1000 images, while small batches comprise 30 images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/cb26d4d2e14c04d96237c32396fee654dc10720f0b6d94dc657d790de21c8f2c.jpg",
+        "image_caption": [
+            "Figure 4: The test accuracy for a range of batch sizes, during training (a) and after 10000 steps (b). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/f3aa0fc16a2e26860bdf9e12234b09eee37b95e9b1f81d398ddde5651cd5ec96.jpg",
+        "image_caption": [
+            "Figure 5: a) The test set accuracy as a function of batch size, for a range of learning rates $\\epsilon$ . The performance peak shifts to the right as we increase $\\epsilon$ , but the overall performance falls once $\\epsilon \\gtrsim 3$ . b) The best observed batch size is proportional to the learning rate across two orders of magnitude. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From now on we focus on the test set accuracy (since this converges as the number of gradient updates increases). In figure 4a, we exhibit training curves for a range of batch sizes between 1 and 1000. We find that the model cannot train when the batch size $B \\lessapprox \\bar { 1 0 }$ . In figure 4b we plot the mean test set accuracy after 10000 training steps. A clear peak emerges, indicating that there is indeed an optimum batch size which maximizes the test accuracy, consistent with Bayesian intuition. The results of Keskar et al. (2016) focused on the decay in test accuracy above this optimum batch size. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "5 STOCHASTIC DIFFERENTIAL EQUATIONS AND THE SCALING RULES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We showed above that the test accuracy peaks at an optimal batch size, if one holds the other SGD hyper-parameters constant. We argued that this peak arises from the tradeoff between depth and breadth in the Bayesian evidence. However it is not the batch size itself which controls this tradeoff, but the underlying scale of random fluctuations in the SGD dynamics. We now identify this SGD “noise scale”, and use it to derive three scaling rules which predict how the optimal batch size depends on the learning rate, training set size and momentum coefficient. A gradient update, ",
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+    {
+        "type": "equation",
+        "img_path": "images/ca2c1619d6fe760545e15462bc4d39b3b0f599e15e0f990417a56ab1927ff7e2.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\Delta \\omega } & { = } & { - \\displaystyle \\frac { \\epsilon } { N } \\left( \\frac { d C } { d \\omega } + \\left( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } \\right) \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/98b0bc094f9ff0d411a45850540a3de33d0a1a5554b328c700b05f4ca887c2a0.jpg",
+        "image_caption": [
+            "Figure 6: a) The test accuracy as a function of batch size, for a range of training set sizes. To reduce noise, we average each curve over five experiments. The performance peak shift to the right as we increase the size of the training set. Unsurprisingly, the overall model performance also improves. b) The best observed batch size is proportional to the size of the training set once $N \\gtrsim 2 0 0 0 \\bar { 0 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            112,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where \u000f is the learning rate, N the training set size, dCdω $\\begin{array} { r } { \\frac { d C } { d \\omega } = \\sum _ { i = 1 } ^ { N } \\frac { d C _ { i } } { d \\omega } } \\end{array}$ the true gradient, and $\\begin{array} { r } { \\frac { d \\hat { C } } { d \\omega } = } \\end{array}$ NB PBi=1 dCidω the estimated gradient evaluated on a mini-batch. The expected gradient of a single example, $\\begin{array} { r } { \\left. \\frac { d C _ { i } } { d \\omega } \\right. = \\frac { 1 } { N } \\frac { d C } { d \\omega } } \\end{array}$ , while $\\begin{array} { r } { \\left. \\frac { d C _ { i } } { d \\omega } \\frac { d C _ { j } } { d \\omega } \\right. = \\left( \\frac { 1 } { N } \\frac { d C } { d \\omega } \\right) ^ { 2 } + F ( \\omega ) \\delta _ { i j } . \\mathrm { ~ } F ( \\omega } \\end{array}$ is a matrix describing the gradient covariances, which are a functheorem and model the gradient error $\\begin{array} { r } { \\alpha = ( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } ) } \\end{array}$ nt par with alues. n rando approximation briefly in appendix C). It is easy to show that $\\langle \\alpha \\rangle = 0$ $\\begin{array} { r } { \\langle \\alpha ^ { 2 } \\rangle = N ( \\frac { N } { B } - 1 ) F ( \\omega ) } \\end{array}$ . Typically $N \\gg B$ , such that $\\langle \\alpha ^ { 2 } \\rangle \\approx N ^ { 2 } F ( \\omega ) / B$ . To continue, we interpret equation 11 as the discrete update of a stochastic differential equation (Li et al., 2017; Gardiner, 1985), ",
+        "bbox": [
+            173,
+            381,
+            825,
+            518
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8158d684a1a27720bd6009bc60d70996048f1cedc99bee78a4e04e9ca506dc5c.jpg",
+        "text": "$$\n\\frac { d \\omega } { d t } = - \\frac { d C } { d \\omega } + \\eta ( t ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            522,
+            566,
+            553
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Where $t$ is a continuous variable, $\\eta ( t )$ represents noise, $\\langle \\eta ( t ) \\rangle = 0$ and $\\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = g F ( \\omega ) \\delta ( t { - } t ^ { \\prime } )$ . The constant $g$ controls the scale of random fluctuations in the dynamics. To relate this differential equation to the SGD, we compute a gradient update $\\begin{array} { r } { \\Delta \\omega = \\int _ { 0 . } ^ { \\epsilon / N } \\frac { d \\omega } { d t } d t = - \\frac { \\epsilon } { N } \\frac { d C } { d \\omega } + \\int _ { 0 } ^ { \\epsilon / N } \\eta ( t ) d t } \\end{array}$ Finally, to measure $\\mathbf { g }$ , we equate the variance in this gradient update to the variance in equation 11, ",
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fb3ddc24790224f84790dc706c3ab25786d957d852bd27c1a3eb6ed8c54924b1.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\left( \\frac { \\epsilon } { N } \\right) ^ { 2 } \\langle \\alpha ^ { 2 } \\rangle } & { = } & { \\displaystyle \\epsilon ^ { 2 } ( \\frac { N } { B } - 1 ) F ( \\omega ) / N } \\\\ { \\displaystyle } & { = } & { \\displaystyle \\langle ( \\int _ { 0 } ^ { \\epsilon / N } d t \\eta ( t ) ) ^ { 2 } \\rangle = \\int _ { 0 } ^ { \\epsilon / N } d t \\int _ { 0 } ^ { \\epsilon / N } d t ^ { \\prime } \\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = \\epsilon g F ( \\omega ) / N . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Rearranging, the SGD noise scale $\\begin{array} { r } { g = \\epsilon ( \\frac { N } { B } - 1 ) \\approx \\epsilon N / B } \\end{array}$ . The noise scale falls when the batch size increases, consistent with our earlier observation of an optimal batch size $B _ { o p t }$ while holding the other hyper-parameters fixed. Notice that one would equivalently observe an optimal learning rate if one held the batch size constant. A similar analysis of the SGD was recently performed by Mandt et al. (2017), although their treatment only holds near local minima where the covariances $F ( \\omega )$ are stationary. Our analysis holds throughout training, which is necessary since Keskar et al. (2016) found that the beneficial influence of noise was most pronounced at the start of training. ",
+        "bbox": [
+            173,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "When we vary the learning rate or the training set size, we should keep the noise scale fixed, which implies that $B _ { o p t } \\propto \\epsilon N$ . In figure 5a, we plot the test accuracy as a function of batch size after $( 1 0 0 0 0 / \\epsilon )$ training steps, for a range of learning rates. Exactly as predicted, the peak moves to the right as $\\epsilon$ increases. Additionally, the peak test accuracy achieved at a given learning rate does not begin to fall until $\\epsilon \\sim 3$ , indicating that there is no significant discretization error in integrating the stochastic differential equation below this point. Above this point, the discretization error begins to dominate and the peak test accuracy falls rapidly. In figure 5b, we plot the best observed batch size as a function of learning rate, observing a clear linear trend, $B _ { o p t } \\propto \\epsilon$ . The error bars indicate the distance from the best observed batch size to the next batch size sampled in our experiments. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/f65c1d948657f9b8bf2cf5faeff5b3749c8f5c3f8c2d509c3f217f125dee5dd5.jpg",
+        "image_caption": [
+            "Figure 7: a) The test set accuracy as a function of batch size for a range of momentum coefficients. As expected, the peak moves to the right as the momentum coefficient increases. b) The best observed batch size for a range of momentum coefficients. The green curve exhibits the scaling rule. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This scaling rule allows us to increase the learning rate with no loss in test accuracy and no increase in computational cost, simply by simultaneously increasing the batch size. We can then exploit increased parallelism across multiple GPUs, reducing model training times (Goyal et al., 2017). A similar scaling rule was independently proposed by Jastrzebski et al. (2017) and Chaudhari & Soatto (2017), although neither work identifies the existence of an optimal noise scale. A number of authors have proposed adjusting the batch size adaptively during training (Friedlander & Schmidt, 2012; Byrd et al., 2012; De et al., 2017), while Balles et al. (2016) proposed linearly coupling the learning rate and batch size within this framework. In Smith et al. (2017), we show empirically that decaying the learning rate during training and increasing the batch size during training are equivalent. ",
+        "bbox": [
+            174,
+            376,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In figure 6a we exhibit the test set accuracy as a function of batch size, for a range of training set sizes after 10000 steps $\\mathbf { \\epsilon } \\cdot \\mathbf { \\epsilon } = 1$ everywhere). Once again, the peak shifts right as the training set size rises, although the generalization gap becomes less pronounced as the training set size increases. In figure 6b, we plot the best observed batch size as a function of training set size; observing another linear trend, $B _ { o p t } \\propto N$ . This scaling rule could be applied to production models, progressively growing the batch size as new training data is collected. We expect production datasets to grow considerably over time, and consequently large batch training is likely to become increasingly common. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, in appendix $\\mathrm { D }$ we extend our analysis to SGD with momentum, identifying the noise scale, $\\begin{array} { r } { g \\approx \\frac { \\epsilon N ^ { \\mathrm { ~ \\scriptsize ~ \\cdots ~ } } } { B ( 1 - m ) } } \\end{array}$ , where $m$ denotes the momentum coefficient. Notice that this reduces to the noise scale of conventional SGD as $m  0$ . When $m > 0$ , we obtain an additional scaling rule $B _ { o p t } \\propto$ $1 / ( 1 - m )$ . This scaling rule predicts that the optimal batch size will increase when the momentum coefficient is increased. In figure 7a we plot the test set performance as a function of batch size after 10000 gradient updates ( $\\mathbf { \\epsilon } _ { \\epsilon } = 1$ everywhere), for a range of momentum coefficients. In figure 7b, we plot the best observed batch size as a function of the momentum coefficient, and fit our results to the scaling rule above; obtaining remarkably good agreement. We propose a simple heuristic for tuning the batch size, learning rate and momentum coefficient in appendix E. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            765,
+            328,
+            780
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Just like deep neural networks, linear models which generalize well on informative labels can memorize random labels of the same inputs. These observations are explained by the Bayesian evidence, which is composed of the cost function and an “Occam factor”. The Occam factor penalizes sharp minima but it is invariant to changes in model parameterization. Mini-batch noise drives SGD away from sharp minima, and therefore there is an optimum batch size which maximizes the test accuracy. Interpreting SGD as the discretization of a stochastic differential equation, we predict this optimum batch size should scale linearly with both the learning rate and the training set size, $B _ { o p t } \\propto \\epsilon N$ . We derive an additional scaling rule, $B _ { o p t } \\propto 1 / ( 1 - \\stackrel { \\textstyle - } { m } )$ , between the optimal batch size and the momentum coefficient. We verify these scaling rules empirically and discuss their implications. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
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+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Pieter-Jan Kindermans, Prajit Ramachandran, Jascha Sohl-Dickstein, Jon Shlens, Kevin Murphy, Samy Bengio, Yasaman Bahri and Saeed Saremi for helpful comments on the manuscript. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A BAYESIAN POSTERIOR SAMPLING AND LANGEVIN DYNAMICS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            541,
+            720,
+            558
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+    },
+    {
+        "type": "text",
+        "text": "Instead of minimizing the cost function, Bayesian usually prefer to sample parameter values from the posterior (MacKay, 1992), ",
+        "bbox": [
+            173,
+            571,
+            825,
+            602
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25ecff15d1d1ee78f02b54607b9bb88464578a2200b1baf5b139a154417faa00.jpg",
+        "text": "$$\nP ( \\omega | y , x ; M ) \\propto e ^ { - C ( \\omega ; M ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            608,
+            591,
+            627
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $C ( \\omega ; M )$ is the regularized summed cost function, as shown in section 2 of the main text. It is well known that one can sample this posterior by simulating the overdamped Langevin equation (Gardiner, 1985), which is described by the stochastic differential equation, ",
+        "bbox": [
+            174,
+            633,
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+            676
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a64a4e88036d9e419e523788d5b40160e27a75b0e22726c6891189a4d9c36c4.jpg",
+        "text": "$$\n\\frac { d \\omega } { d t } = - \\frac { d C } { d \\omega } + \\eta ( t ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            683,
+            566,
+            714
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $t$ is a continuous variable, and $\\eta ( t )$ describes Gaussian noise with mean $\\langle \\eta ( t ) \\rangle = 0$ and variance $\\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = 2 T I \\delta ( t - t ^ { \\prime } )$ . The matrix $I$ denotes the identity, while $T$ is the “temperature”. Notice this Langevin equation is extremely similar to the stochastic differential equation of SGD, discussed in section 5 of the main text. Indeed, if the gradient covariances $F ( \\omega )$ were stationary and proportional to the identity, then the SGD would integrate an overdamped Langevin equation with temperature proportional to the SGD noise scale $g$ . As $t \\to \\infty$ , the probability of sampling any particular parameter vector $\\omega$ from the Langevin equation, $P ( \\omega , t  \\infty ) \\propto e ^ { - C / \\dot { T } }$ . ",
+        "bbox": [
+            173,
+            719,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We obtain posterior samples if $T = 1$ . In order to draw posterior samples in practice, we repeatedly integrate the Langevin equation (at temperature $T = 1$ ), over a finite step $t \\to t + \\epsilon / N$ , ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/297d5a317e1bfec16b10ffb8e7983fb04010fb2f576126cd6c90a74727bb0ef1.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\Delta \\omega } & { = } & { - \\displaystyle \\frac { \\epsilon } { N } \\frac { d C } { d \\omega } + \\int _ { t } ^ { t + \\frac { \\epsilon } { N } } \\eta ( t ) d t } \\\\ & { = } & { \\displaystyle - \\frac { \\epsilon } { N } \\frac { d C } { d \\omega } + \\alpha , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            861,
+            620,
+            933
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha$ denotes a Gaussian random variable with mean $\\langle \\alpha \\rangle = 0$ and variance $\\langle \\alpha ^ { 2 } \\rangle \\ : = \\ : 2 \\epsilon I / N$ , which introduces isotropic noise to the gradient update as described in section 4 of the main text. Note that, since $C ( \\omega ; M )$ denotes the summed cost function, we chose to scale our step size by the training set size $N$ . This also matches our treatment of SGD in section 5 of the main text. The larger the step size $\\epsilon$ , the greater the discretization error, but if $\\epsilon$ is sufficiently small and we iterate equation 17 sufficiently many times, we will obtain valid samples from the posterior. ",
+        "bbox": [
+            174,
+            102,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Since the probability of sampling any given parameter vector $\\omega$ is proportional to the posterior, the probability of sampling a parameter vector belonging to any given local minimum is proportional to the integral of the posterior over the bowl of attraction $D$ which surrounds that minimum. ",
+        "bbox": [
+            174,
+            194,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a67abd61022557b8527f0ceadb80a46befc91282e1cd1c40bfff4965dfd908a0.jpg",
+        "text": "$$\nP ( \\omega \\in D , t  \\infty ) \\propto \\int _ { D } d \\omega \\ e ^ { - C ( \\omega ; M ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            239,
+            632,
+            273
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Meanwhile we showed in section 2 of the main text that the evidence in favor of a model is proportional to the integral of the posterior over all parameter space. ",
+        "bbox": [
+            176,
+            275,
+            823,
+            304
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/645eab6b826d2ff9c83f869bf1f7b717a71bcdf29443928aef6d566e2f195ad3.jpg",
+        "text": "$$\nP ( y | x ; M ) \\propto \\int d \\omega e ^ { - C ( \\omega ; M ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            306,
+            606,
+            339
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "As we discussed, this evidence is dominated by the contributions to the integral near local minima. In a convex model, there is only one such minimum; which allows us to accurately estimate the model evidence. Meanwhile, in non-convex models, there are many such minima, and so we can instead define the evidence as a sum over local evidences in favor of each minimum, ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d21a0bc2f36dbe12c5a5f6c1e18aa3112145d4a431dcd517692d713c252e91b.jpg",
+        "text": "$$\nP ( y | x ; M ) = \\sum _ { i } P ( y | x , \\omega \\in D _ { i } ; M ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            401,
+            627,
+            434
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where we define the evidence in favor of a minimum as the integral over the local bowl of attraction, ",
+        "bbox": [
+            174,
+            436,
+            821,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1585219cdde94fb4528524bae5b3410dec703bc1860283c24f667d7be7f7137f.jpg",
+        "text": "$$\nP ( y | x , \\omega \\in D ; M ) \\propto \\int _ { D } d \\omega \\ e ^ { - C ( \\omega ; M ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            460,
+            633,
+            493
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Since the combined bowls of attraction of all the minima perfectly tile the entire parameter space, equations 19 and 20 are equivalent. Meanwhile, equating equations 18 and 21 we find that, when one performs Bayesian posterior sampling, the probability of sampling the parameter vector from a local minimum is proportional to the evidence in favor of that minimum. This demonstrates that bayesian posterior samples are biased in favor of local minima whose evidence is large, which explains why a single posterior sample $\\omega _ { p }$ often achieves lower test error than the cost function minimum $\\omega _ { 0 }$ . ",
+        "bbox": [
+            173,
+            493,
+            825,
+            578
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B THE EFFECT OF REGULARIZATION ON THE GENERALIZATION GAP ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            598,
+            751,
+            613
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In the experiments of section 4 of the main text, the L2 regularization coefficient $\\lambda = 0$ . In figure 8, we plot the evolution of the training curves when $\\lambda = 0 . 1$ , for both small batch and full batch training. Excluding the regularization parameter, these experiments are identical to figure 3. To our surprise, regularized full batch training took longer to converge than small batch training. In another surprise, regularization significantly reduced the size of the generalization gap. While large batch regularized training achieves slightly lower test set accuracy than unregularized small batch training, it also achieves lower test cross-entropy. The test cross-entropy of our regularized models does not degrade after many gradient updates, removing the need for early stopping. ",
+        "bbox": [
+            171,
+            627,
+            825,
+            656
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b725df1b2851f06eebf187dcc70cd8ce8c558a8b2aed25f92d9127ad2a42bfe.jpg",
+        "image_caption": [
+            "Figure 8: The mean test accuracy (a) and the mean test cross-entropy (b) of a regularized model during training. While full batch training takes longer to converge, it achieves similar performance at long times. The noise inherent in small batch training causes the performance to fluctuate. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            688,
+            803,
+            867
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e9f4bc9e0527e3032e6c0929c574be057d50f2e1d13b90d5cc6350b4fcdf353.jpg",
+        "image_caption": [
+            "Figure 9: The gradient distribution of a randomly selected parameter in the softmax layer, when measured over a single training example (a), and when averaged over mini-batches of 30 images (b). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            106,
+            808,
+            286
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            353,
+            825,
+            438
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C THE GAUSSIAN APPROXIMATION TO THE MINI-BATCH ERROR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            715,
+            476
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In section 5 of the main text, we approximated the difference between the full batch gradient and the mini-batch gradient estimate, $\\begin{array} { r } { \\alpha = ( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } ) } \\end{array}$ , by a Gaussian random variable. This enabled us to derive the scaling rules, which we verified empirically. We motivated this assumption by reference to the central limit theorem, which states that the gradient error will tend towards Gaussian noise as $\\{ N \\to \\infty , B \\to \\infty , B \\ll N \\}$ , so long as the distribution of gradients over individual training examples does not have heavy tails. In practice neither $\\mathbf { N }$ nor B is infinite, and the gradient distribution may be heavy tailed, especially when gradients are sparse. Nonetheless the central limit theorem tends to be surprisingly robust in practice, and is consequently widely used. ",
+        "bbox": [
+            174,
+            491,
+            825,
+            607
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "It is beyond the scope of this work to perform a thorough study of the gradient noise distribution in deep networks. However as a brief proof of principle, we present the distribution of the gradient immediately after random initialization in figure 9, for the shallow neural network discussed in sections 4 and 5 of the main text. In figure 9a, we present the distribution over the individual training examples, of the gradient of a single matrix element in the softmax output layer, chosen randomly. The distribution is double peaked and clearly not Gaussian. However in figure 7b, we plot the distribution of the gradient of the same matrix element, when averaged over randomly sampled mini-batches of 30 images (without replacement). A single peak emerges, and while the distribution is still slightly skewed, it is clearly already approaching the Gaussian limit. We conclude that the Gaussian approximation is likely to be reasonable for commonly used mini-batch sizes. ",
+        "bbox": [
+            173,
+            613,
+            825,
+            752
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D DERIVING THE SCALING RULES FOR SGD WITH MOMENTUM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            773,
+            714,
+            790
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Momentum simulates a generalized Langevin equation (with structured fluctuations), ",
+        "bbox": [
+            174,
+            805,
+            728,
+            820
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/16e299b6d82ae6a50ef8986f4a0a56d4aff5be1686a4f726b75f607a014315fa.jpg",
+        "text": "$$\n\\frac { d ^ { 2 } \\omega } { d t ^ { 2 } } = - \\lambda \\frac { d \\omega } { d t } - \\frac { d C } { d \\omega } + \\eta ( t ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            828,
+            598,
+            861
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "$\\lambda$ is the “damping coefficient” and $\\eta ( t )$ describes Gaussian noise, whose statistics $\\langle \\eta ( t ) \\rangle = 0$ and $\\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = \\bar { g } \\lambda \\bar { F } ( w ) \\delta ( t - t ^ { \\prime } )$ . As before, the coefficient $g$ describes the scale of random fluctuations in the dynamics, and $F ( \\omega )$ describes the gradient covariances between parameters. We include a factor of $\\lambda$ in the noise variance to satisfy the fluctuation-dissipation theorem, which states that we can vary the damping coefficient without changing the probability of sampling any particular configuration of parameters in the limit $t \\to \\infty$ , if we proportionally increase the noise variance. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To relate this Langevin equation to the usual momentum equations, we first re-express it as two coupled first order differential equations, ",
+        "bbox": [
+            174,
+            138,
+            821,
+            167
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14b1af7cbf0612e821a8578c052eb4002b59937c43eb316113a8b9a7f75d56c3.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\frac { d p } { d t } } & { = } & { \\displaystyle - \\lambda p - \\frac { d C } { d \\omega } + \\eta ( t ) , } \\\\ { \\displaystyle \\frac { d \\omega } { d t } } & { = } & { p . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            169,
+            596,
+            232
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Integrating over a single step $\\Delta t / N$ , ",
+        "bbox": [
+            173,
+            233,
+            416,
+            248
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/527e2c6b8556aa2e9cfdf1a776a8ed658b8aa5b27d1699cdc2818fec6eb54a4c.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\Delta p } } & { { = } } & { { \\displaystyle - ( \\lambda \\Delta t / N ) p - \\frac { \\Delta t } { N } \\frac { d C } { d \\omega } + \\eta , } } \\\\ { { \\Delta \\omega } } & { { = } } & { { p \\Delta t / N . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            250,
+            627,
+            299
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Where now $\\langle \\eta \\rangle = 0$ and $\\langle \\eta ^ { 2 } \\rangle = g \\Delta t \\lambda F ( w ) / N$ . We define the accumulation $A = p / \\Delta t$ , ",
+        "bbox": [
+            173,
+            300,
+            754,
+            316
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2524bb54c465e3bf98a6c473a33cef667411bfca45d9b50835703c85a98557d.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\Delta A } & { = } & { - ( \\lambda \\Delta t / N ) A - \\displaystyle \\frac { 1 } { N } \\frac { d C } { d \\omega } + \\frac { \\eta } { \\Delta t } , } \\\\ { \\Delta \\omega } & { = } & { ( \\Delta t ) ^ { 2 } A / N . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            318,
+            633,
+            368
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "These equations can be compared to the TensorFlow update equations for momentum, ",
+        "bbox": [
+            174,
+            369,
+            736,
+            385
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb7199cb470e8913109de33649db9c87f6bd9c85ffff3aeff0506100287f699a.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\Delta A } } & { { = } } & { { ( m - 1 ) A - \\displaystyle \\frac { 1 } { N } \\left( \\displaystyle \\frac { d C } { d \\omega } + \\alpha \\right) , } } \\\\ { { \\Delta \\omega } } & { { = } } & { { \\epsilon A . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            387,
+            632,
+            439
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Where $\\begin{array} { r } { \\alpha = \\left( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } \\right) } \\end{array}$ denotes the error in the gradient update. As discussed in the main text, we can approximate this error as Gaussian noise with statistics $\\langle \\alpha \\rangle = 0$ and $\\langle \\alpha ^ { 2 } \\rangle \\approx N ^ { 2 } F ( \\omega ) / B$ . Equations 27 and 28 match equations 29 and 30 if the step size $\\dot { \\epsilon } = ( \\Delta t ) ^ { 2 } / N$ , and the momentum parameter $m = ( 1 - \\lambda \\Delta t / N )$ . Finally we equate the noise by setting $\\langle \\alpha ^ { 2 } \\rangle / N ^ { 2 } = \\langle \\eta ^ { 2 } \\rangle / ( \\Delta t ) ^ { 2 }$ , and solve for the noise scale $g$ to obtain, ",
+        "bbox": [
+            174,
+            441,
+            825,
+            520
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1250fa8ad6b646385b1e82ba5b00b1623c3499d996f37b0e6dd9e4e343faa1b5.jpg",
+        "text": "$$\n\\begin{array} { l l l } { { g } } & { { \\approx } } & { { \\displaystyle \\frac { \\Delta t N } { B \\lambda } = \\frac { ( \\Delta t ) ^ { 2 } } { B ( 1 - m ) } } } \\\\ { { } } & { { = } } & { { \\displaystyle \\frac { \\epsilon N } { B ( 1 - m ) } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            522,
+            593,
+            593
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As observed in the main text, if we wish to keep the scale of random fluctuations constant, then we should scale the batch size $B \\propto \\epsilon N$ . We also predict an additional scaling relation between the batch size and the momentum parameter, $B \\propto \\mathsf { \\bar { 1 } } / ( 1 - m )$ . Note that one can also interpret $\\epsilon _ { e f f } = \\epsilon / ( 1 - m )$ as the “effective learning rate”. ",
+        "bbox": [
+            174,
+            593,
+            825,
+            650
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E HOW TO ACHIEVE LARGE BATCH TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            669,
+            568,
+            684
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here we propose a simple heuristic for tuning the batch size, learning rate and momentum parameter; in order to maximize both test accuracy and batch size (enabling parallel training across many machines). Note that this is only worthwhile if one expects to retrain a model many times. ",
+        "bbox": [
+            174,
+            698,
+            825,
+            741
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. Set the learning rate to 0.1 and the momentum coefficient to 0.9. Run experiments at a range of batch sizes on a logarithmic scale, and identify the optimal batch size which maximizes the validation set accuracy. If training is not stable, reduce the learning rate, and repeat. 2. Repeatedly increase the batch size by a factor of 3, while scaling the learning rate $\\epsilon \\propto B$ , until the validation set accuracy starts to fall. Then repeatedly increase the batch size by a factor of 3, while scaling the momentum coefficient $( 1 - \\dot { m } ) \\propto 1 / B$ , until either the validation set accuracy falls or the batch size reaches the limits of your hardware. 3. Having identified the final learning rate and momentum parameter, retune the batch size on a linear scale in the local neighborhood of the current batch size. ",
+        "bbox": [
+            210,
+            752,
+            825,
+            886
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We believe that this simple procedure will increase the test accuracy, reduce the cost of tuning hyperparameters, and significantly reduce the final number of gradient updates required to train a model. ",
+        "bbox": [
+            174,
+            895,
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+        ],
+        "page_idx": 12
+    }
+]

parse/train/Byl1W1rtvH/Byl1W1rtvH.md

@@ -0,0 +1,376 @@
+# RECURRENT HIERARCHICAL TOPIC-GUIDEDNEURAL LANGUAGE MODELS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+To simultaneously capture syntax and global semantics from a text corpus, we propose a new larger-context recurrent neural network (RNN) based language model, which extracts recurrent hierarchical semantic structure via a dynamic deep topic model to guide natural language generation. Moving beyond a conventional RNN based language model that ignores long-range word dependencies and sentence order, the proposed model captures not only intra-sentence word dependencies, but also temporal transitions between sentences and inter-sentence topic dependences. For inference, we develop a hybrid of stochastic-gradient MCMC and recurrent autoencoding variational Bayes. Experimental results on a variety of real-world text corpora demonstrate that the proposed model not only outperforms state-ofthe-art larger-context RNN-based language models, but also learns interpretable recurrent multilayer topics and generates diverse sentences and paragraphs that are syntactically correct and semantically coherent.
+
+# 1 INTRODUCTION
+
+Both topic and language models are widely used for text analysis. Topic models, such as latent Dirichlet allocation (LDA) (Blei et al., 2003; Griffiths & Steyvers, 2004; Hoffman et al., 2013) and its nonparametric Bayesian generalizations (Teh et al., 2006; Zhou & Carin, 2015), are well suited to extract document-level word concurrence patterns into latent topics from a text corpus. Their modeling power has been further enhanced by introducing multilayer deep representation (Srivastava et al., 2013; Mnih & Gregor, 2014; Gan et al., 2015; Zhou et al., 2016; Zhao et al., 2018; Zhang et al., 2018). While having semantically meaningful latent representation, they typically treat each document as a bag of words (BoW), ignoring word order (Griffiths et al., 2004; Wallach, 2006). Language models have become key components of various natural language processing (NLP) tasks, such as text summarization (Rush et al., 2015; Gehrmann et al., 2018), speech recognition (Mikolov et al., 2010; Graves et al., 2013), machine translation (Sutskever et al., 2014; Cho et al., 2014), and image captioning (Vinyals et al., 2015; Mao et al., 2015; Xu et al., 2015; Gan et al., 2017; Rennie et al., 2017). The primary purpose of a language model is to capture the distribution of a word sequence, commonly with a recurrent neural network (RNN) (Mikolov et al., 2011; Graves, 2013) or a Transformer based neural network (Vaswani et al., 2017; Dai et al., 2019; Devlin et al., 2019; Radford et al., 2018; 2019). In this paper, we focus on improving RNN-based language models that often have much fewer parameters and are easier to perform end-to-end training.
+
+While RNN-based language models do not ignore word order, they often assume that the sentences of a document are independent to each other. This simplifies the modeling task to independently assigning probabilities to individual sentences, ignoring their orders and document context (Tian & Cho, 2016). Such language models may consequently fail to capture the long-range dependencies and global semantic meaning of a document (Dieng et al., 2017; Wang et al., 2018). To relax the sentence independence assumption in language modeling, Tian & Cho (2016) propose larger-context language models that model the context of a sentence by representing its preceding sentences as either a single or a sequence of BoW vectors, which are then fed directly into the sentence modeling RNN. An alternative approach attracting significant recent interest is leveraging topic models to improve RNN-based language models. Mikolov & Zweig (2012) use pre-trained topic model features as an additional input to the RNN hidden states and/or output. Dieng et al. (2017); Ahn et al. (2017) combine the predicted word distributions, given by both a topic model and a language model, under variational autoencoder (Kingma & Welling, 2013). Lau et al. (2017) introduce an attention based convolutional neural network to extract semantic topics, which are used to extend the RNN cell. Wang et al. (2018) learn the global semantic coherence of a document via a neural topic model and use the learned latent topics to build a mixture-of-experts language model. Wang et al. (2019) further specify a Gaussian mixture model as the prior of the latent code in variational autoencoder, where each mixture component corresponds to a topic.
+
+While clearly improving the performance of the end task, these existing topic-guided methods still have clear limitations. For example, they only utilize shallow topic models with only a single stochastic hidden layer in their data generation process. Note several neural topic models use deep neural networks to construct their variational encoders, but still use shallow generative models (decoders) (Miao et al., 2017; Srivastava & Sutton, 2017). Another key limitation lies in ignoring the sentence order, as they treat each document as a bag of sentences. Thus once the topic weight vector learned from the document context is given, the task is often reduced to independently assigning probabilities to individual sentences (Lau et al., 2017; Wang et al., 2018; 2019).
+
+In this paper, as depicted in Fig. 1, we propose to use recurrent gamma belief network (rGBN) to guide a stacked RNN for language modeling. We refer to the model as rGBN-RNN, which integrates rGBN (Guo et al., 2018), a deep recurrent topic model, and stacked RNN (Graves, 2013; Chung et al., 2017), a neural language model, into a novel larger-context RNN-based language model. It simultaneously learns a deep recurrent topic model, extracting document-level multi-layer word concurrence patterns and sequential topic weight vectors for sentences, and an expressive language model, capturing both short- and long-range word sequential dependencies. For inference, we equip rGBN-RNN (decoder) with a novel variational recurrent inference network (encoder), and train it end-to-end by maximizing the evidence lower bound (ELBO). Different from the stacked RNN based language model in Chung et al. (2017), which relies on three types of customized training operations (UPDATE, COPY, FLUSH) to extract multi-scale structures, the language model in rGBN-RNN learns such structures purely under the guidance of the temporally and hierarchically connected stochastic layers of rGBN. The effectiveness of rGBN-RNN as a new larger-context language model is demonstrated both quantitatively, with perplexity and BLEU scores, and qualitatively, with interpretable latent structures and randomly generated sentences and paragraphs. Notably, rGBN-RNN can generate a paragraph consisting of a sequence of semantically coherent sentences.
+
+# 2 RECURRENT HIERARCHICAL TOPIC-GUIDED LANGUAGE MODEL
+
+Denote a document of $J$ sentences as $\mathcal { D } = ( S _ { 1 } , S _ { 2 } , \ldots , S _ { J } )$ , where $S _ { j } = ( y _ { j , 1 } , \ldots , y _ { j , T _ { j } } )$ consists of $T _ { j }$ words from a vocabulary of size $V$ . Conventional statistical language models often only focus on the word sequence within a sentence. Assuming that the sentences of a document are independent to each other, they often define $\begin{array} { r } { P ( \mathcal { D } ) \approx \prod _ { j = 1 } ^ { J } \bar { P } \left( S _ { j } \right) = \prod _ { j = 1 } ^ { J } \prod _ { t = 2 } ^ { T _ { j } } p \left( y _ { j , t } \vert y _ { j , < t } \right) p \left( y _ { j , 1 } \right) } \end{array}$ . RNN based neural language models define the conditional probability of each word $y _ { j , t }$ given all the previous words $y _ { j , < t }$ within the sentence $S _ { j }$ , through the softmax function of a hidden state $h _ { j , t }$ , as
+
+$$
+p \left( y _ { j , t } \mid y _ { j , < t } \right) = p \left( y _ { j , t } \mid h _ { j , t } \right) , \quad h _ { j , t } = f \left( h _ { j , < t } , y _ { j , t - 1 } \right) ,
+$$
+
+where $f ( \cdot )$ is a non-linear function typically defined as an RNN cell, such as long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997) and gated recurrent unit (GRU) (Cho et al., 2014).
+
+These RNN-based statistical language models are typically applied only at the word level, without exploiting the document context, and hence often fail to capture long-range dependencies. While Dieng et al. (2017); Lau et al. (2017); Wang et al. (2018; 2019) remedy the issue by guiding the language model with a topic model, they still treat a document as a bag of sentences, ignoring the order of sentences, and lack the ability to extract hierarchical and recurrent topic structures.
+
+We introduce rGBN-RNN, as depicted in Fig. 1(a), as a new larger-context language model. It consists of two key components: (i) a hierarchical recurrent topic model (rGBN), and (ii) a stacked RNN based language model. We use rGBN to capture both global semantics across documents and long-range inter-sentence dependencies within a document, and use the language model to learn the local syntactic relationships between the words within a sentence. Similar to Lau et al. (2017); Wang et al. (2018), we represent a document as a sequence of sentence-context pairs as $( \{ S _ { 1 } , d _ { 1 } \} , \dots , \{ S _ { J } , d _ { J } \} )$ , where $\bar { d } _ { j } \in \mathbb { Z } _ { + } ^ { V _ { c } }$ summarizes the document excluding $S _ { j }$ , specifically $( S _ { 1 } , . . . , S _ { j - 1 } , S _ { j + 1 } , . . . , S _ { J } )$ , into a BoW count vector, with $V _ { c }$ as the size of the vocabulary excluding stop words. Note a naive way is to treat each sentence as a document, use a dynamic topic model (Blei & Lafferty, 2006) to capture the temporal dependencies of the latent topic-weight vectors, which is fed to the RNN to model the word sequence of the corresponding sentence. However, the sentences are often too short to be well modeled by a topic model. In our setting, as $d _ { j }$ summarizes the document-level context of $S _ { j }$ , it is in general sufficiently long for topic modeling. Note during testing, we redefine $d _ { j }$ as the BoW vector summarizing only the preceding sentences, i.e., $S _ { 1 : j - 1 }$ , which will be further clarified when presenting experimental results.
+
+![](images/282e53503b935c73b464c066fccb5b2394fe7d7bb548b09489edae9d737bf127.jpg)  
+Figure 1: (a) The generative model of a three-hidden-layer rGBN-RNN, where the bottom part is the deep recurrent topic model (rGBN), document contexts of consecutive sentences are used as observed data, and upper is the language model. (b) Overview of the language model component, where input $x _ { j , t }$ denotes the tth word in $j$ th sentence of a document, $x _ { j , t } = y _ { j , t - 1 } , h _ { j , t } ^ { l }$ is the hidden state of the stacked RNN at time step $t$ , and $\theta _ { j } ^ { l }$ is the topic weight vector of sentence $j$ at layer $l$ . (c) The overall architecture of the proposed model, including the decoder (rGBN and language model) and encoder (variational recurrent inference), where the red arrows denote the inference of latent topic weight vectors, black ones the data generation.
+
+# 2.1 HIERARCHICAL RECURRENT TOPIC MODEL
+
+Shown in Fig. 1 (a), to model the time-varying sentence-context count vectors $d _ { j }$ in document $\mathcal { D }$ , the generative process of the rGBN component, from the top to bottom hidden layers, is expressed as
+
+$$
+\begin{array} { r l } & { \pmb { \theta } _ { j } ^ { L } \sim \mathrm { G a m } \left( \mathbf { \Pi } \mathbf { \ } \mathbf { \Pi } ^ { L } \pmb { \theta } _ { j - 1 } ^ { L } , \mathbf { \Delta } \tau _ { 0 } \right) , \cdots , \pmb { \theta } _ { j } ^ { l } \sim \mathrm { G a m } \left( \Phi ^ { l + 1 } \pmb { \theta } _ { j } ^ { l + 1 } + \mathbf { \Pi } \mathbf { \Pi } ^ { l } \pmb { \theta } _ { j - 1 } ^ { l } , \mathbf { \Delta } \tau _ { 0 } \right) , \cdots , } \\ & { \pmb { \theta } _ { j } ^ { 1 } \sim \mathrm { G a m } \left( \Phi ^ { 2 } \pmb { \theta } _ { j } ^ { 2 } + \mathbf { \Pi } \mathbf { \ } ^ { 1 } \pmb { \theta } _ { j - 1 } ^ { 1 } , \mathbf { \Delta } \tau _ { 0 } \right) , \pmb { d } _ { j } \sim \mathrm { P o i s } \left( \Phi ^ { 1 } \pmb { \theta } _ { j } ^ { 1 } \right) , } \end{array}
+$$
+
+where $\pmb { \theta } _ { j } ^ { l } \in \mathbb { R } _ { + } ^ { K _ { l } }$ denotes the gamma distributed topic weight vectors of sentence $j$ at layer $l$ $\mathbf { I I } ^ { l } \in \mathbb { R } _ { + } ^ { \bar { K } _ { l } \times K _ { l } }$ the transition matrix of layer $l$ that captures cross-topic temporal dependencies, $\Phi ^ { l } \in$ RKl−1×Kl+ the loading matrix at layer l, Kl the number of topics of layer l, and τ0 ∈ R+ a scaling hyperparameter. At $j = 1$ , $\pmb { \theta } _ { 1 } ^ { l } \sim \mathrm { G a m } \left( \pmb { \Phi } ^ { l + 1 } \pmb { \theta } _ { 1 } ^ { l + 1 } , \tau _ { 0 } \right)$ for $l = 1 , \ldots , L - 1$ and $\pmb { \theta } _ { 1 } ^ { L } \sim \mathbf { G a m } ( \nu , \tau _ { 0 } )$ , where $\nu = \mathbf { 1 } _ { K _ { L } }$ . Finally, Dirichlet priors are placed on the columns of $\Pi ^ { l }$ and $\Phi ^ { l }$ , i.e., $\pi _ { k } ^ { l }$ and $\phi _ { k } ^ { l }$ , which not only makes the latent representation more identifiable and interpretable, but also facilitates inference. The count vector $d _ { j }$ can be factorized into the product of $\Phi ^ { \hat { 1 } }$ and $\theta _ { j } ^ { 1 }$ under the Poisson likelihood. The shape parameters of $\pmb { \theta } _ { j } ^ { l } \in \mathbb { R } _ { + } ^ { K _ { l } }$ can be factorized into the sum of $\Phi ^ { l + 1 } \pmb { \theta } _ { j } ^ { l + 1 }$ , capturing inter-layer hierarchical dependence, and $\Pi ^ { l } \theta _ { j - 1 } ^ { l }$ , capturing intra-layer temporal dependence. rGBN not only captures the document-level word occurrence patterns inside the training text corpus, but also the sequential dependencies of the sentences inside a document. Note ignoring the recurrent structure, rGBN will reduce to the gamma belief network (GBN) of Zhou et al. (2016), which can be considered as a multi-stochastic-layer deep generalization of LDA (Cong et al., 2017a). If ignoring its hierarchical structure (i.e., $L = 1$ ), rGBN reduces to Poisson–gamma dynamical systems (Schein et al., 2016). We refer to the rGBN-RNN without its recurrent structure as GBN-RNN, which no longer models sequential sentence dependencies; see Appendix A for more details.
+
+# 2.2 LANGUAGE MODEL
+
+Different from a conventional RNN-based language model, which predicts the next word only using the preceding words within the sentence, we integrate the hierarchical recurrent topic weight vectors $\theta _ { j } ^ { l }$ into the language model to predict the word sequence in the $j$ th sentence. Our proposed language model is built upon the stacked RNN proposed in Graves (2013); Chung et al. (2017), but with the help of rGBN, it no longer requires specialized training heuristics to extract multi-scale structures. As shown in Fig. 1 (b), to generate $y _ { j , t }$ , the $t ^ { \mathrm { { t h } } }$ token of sentence $j$ in a document, we construct the hidden states $h _ { j , t } ^ { l }$ of the language model, from the bottom to top layers, as
+
+$$
+\begin{array} { r } { h _ { j , t } ^ { l } = \left\{ \begin{array} { l l } { \mathrm { L S T M } _ { \mathrm { w o r d } } ^ { l } \left( h _ { j , t - 1 } ^ { l } , W _ { e } \left[ x _ { j , t } \right] \right) , } & { \mathrm { i f } l = 1 , } \\ { \mathrm { L S T M } _ { \mathrm { w o r d } } ^ { l } \left( h _ { j , t - 1 } ^ { l } , a _ { j , t } ^ { l - 1 } \right) , } & { \mathrm { i f } L \geq l > 1 , } \end{array} \right. } \end{array}
+$$
+
+where $\mathrm { L S T M } _ { \mathrm { w o r d } } ^ { l }$ denotes the word-level LSTM at layer $l$ , ${ \bf W } _ { e } \in \mathbb { R } ^ { V }$ are word embeddings to be learned, and $x _ { j , t } = y _ { j , t - 1 }$ . Note $\pmb { a } _ { j , t } ^ { l }$ denotes the coupling vector, which combines the temporal topic weight vectors $\theta _ { j } ^ { l }$ and hidden output of the word-level LSTM $h _ { j , t - 1 } ^ { l }$ at each time step $t$ .
+
+Following Lau et al. (2017), a gating unit similar to GRU (Cho et al., 2014) combines $\theta _ { j } ^ { l }$ of sentence $j$ with its hidden state $h _ { j , t } ^ { l }$ of word-level LSTM at layer $l$ and time $t$ as $\mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { }  \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { }  \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf  \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf  \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf $ . We defer the details on $g ^ { l }$ to Appendix B. Denote $\mathbf { W } _ { o }$ as a weight matrix with $V$ rows and $\pmb { a } _ { j , t } ^ { 1 : L }$ as the concatenation of $\boldsymbol { a } _ { j , t } ^ { l }$ across all layers; different from (1), the conditional probability of $y _ { j , t }$ becomes
+
+$$
+p \left( y _ { j , t } \mid y _ { j , < t } , \pmb { \theta } _ { j } ^ { l } \right) = \mathrm { s o f t m a x } \left( \mathbf { W } _ { o } \pmb { a } _ { j , t } ^ { 1 : L } \right) .
+$$
+
+There are two main reasons for combining all the latent representations $\cdot$ for language modeling. First, the latent representations exhibit different statistical properties at different stochastic layers of rGBN-RNN, and hence are combined together to enhance their representation power. Second, having “skip connections” from all hidden layers to the output one makes it easier to train the proposed network, reducing the number of processing steps between the bottom of the network and the top and hence mitigating the “vanishing gradient” problem (Graves, 2013).
+
+To sum up, as depicted in Fig. 1 (a), the topic weight vector $\theta _ { j } ^ { l }$ of sentence $j$ quantifies the topic usage of its document context $d _ { j }$ at layer $l$ . It is further used as an additional feature of the language model to guide the word generation inside sentence $j$ , as shown in Fig. 1 (b). It is clear that rGBN-RNN has two temporal structures: a deep recurrent topic model to extract the temporal topic weight vectors from the sequential document contexts, and a language model to estimate the probability of each sentence given its corresponding hierarchical topic weight vector. Characterizing the word-sentencedocument hierarchy to incorporate both intra- and inter-sentence information, rGBN-RNN learns more coherent and interpretable topics and increases the generative power of the language model. Distinct from existing topic-guided language models, the temporally related hierarchical topics of rGBN exhibit different statistical properties across layers, which better guides language model to improve its language generation ability.
+
+# 2.3 MODEL LIKELIHOOD AND INFERENCE
+
+For rGBN-RNN, given $\{ \Phi ^ { l } , \pmb { \Pi } ^ { l } \} _ { l = 1 } ^ { L }$ , the marginal likelihood of the sequence of sentence-context pairs $( \{ s _ { 1 } , d _ { 1 } \} , \dots , \{ s _ { J } , d _ { J } \} )$ of document $\mathcal { D }$ is defined as
+
+$$
+\begin{array} { r } { P \left( \mathcal { D } \mid \{ \Phi ^ { l } , \mathbf { T } ^ { l } \} _ { l = 1 } ^ { L } \right) = \int \prod _ { j = 1 } ^ { J } \left\{ p \left( d _ { j } \mid \Phi ^ { 1 } \theta _ { j } ^ { 1 } \right) \left[ \prod _ { t = 1 } ^ { T _ { j } } p \left( y _ { j t } \mid y _ { j , < t } , \theta _ { j } ^ { 1 ; L } \right) \right] \left[ \prod _ { l = 1 } ^ { L } p \left( \theta _ { j } ^ { l } \mid e _ { j } ^ { l } , \tau _ { 0 } \right) \right] \right\} d \theta _ { 1 : J } ^ { 1 : L } , } \end{array}
+$$
+
+where $\mathbf { \boldsymbol { e } } _ { j } ^ { l } : = \Phi ^ { l + 1 } \mathbf { \boldsymbol { \theta } } _ { j } ^ { l + 1 } + \Pi ^ { l } \mathbf { \boldsymbol { \theta } } _ { j - 1 } ^ { l }$ . The inference task is to learn the parameters of both the topic model and language model components. One naive solution is to alternate the training between these two components in each iteration: First, the topic model is trained using a sampling based iterative algorithm provided in Guo et al. (2018); Second, the language model is trained with maximum likelihood estimation under a standard cross-entropy loss. While this naive solution can utilize readily available inference algorithms for both rGBN and the language model, it may suffer from stability and convergence issues. Moreover, the need to perform a sampling based iterative algorithm for rGBN inside each iteration limits the scalability of the model for both training and testing.
+
+Algorithm 1 Hybrid SG-MCMC and recurrent autoencoding variational inference for rGBN-RNN.   
+
+<table><tr><td>Set mini-batch size m and the number of layer L Initialize encoder and neural languagemodel parameter parameter Ω,and topic model parameter {Φl,l}=1·</td></tr><tr><td>foriter=1,2,. do</td></tr><tr><td>Randomly select a mini-batch of m documents consisting of J sentences to form a subset X =</td></tr><tr><td>{di,1:J,Si,1:J}m1; m,J,L</td></tr><tr><td>Draw random noise {e&#x27;,j }i=1.j=1,l=1 from uniform distribution;</td></tr><tr><td>Calculate VΩL (S2, Φl,II&#x27;; X,e,j) according to (6),and update Ω; Sample 0’,j from (7) and (8) via Ω</td></tr><tr><td>to update {II′}=1 and {Φ′}𝑖=1, will be described in Appendix C; end for</td></tr></table>
+
+To this end, we introduce a variational recurrent inference network (encoder) to learn the latent temporal topic weight vectors $\cdot$ . Denoting $\begin{array} { r } { Q = \prod _ { j = 1 } ^ { J } \prod _ { l = 1 } ^ { L } q ( \pmb { \theta } _ { j } ^ { l } | \pmb { d } _ { \leq j } ) } \end{array}$ , the ELBO of the log marginal likelihood shown in (5) can be constructed as
+
+$$
+\begin{array} { r } { L = \sum _ { j = 1 } ^ { J } \mathbb { E } _ { Q } \left[ \ln p \left( d _ { j } | \Phi | ^ { \mathbf { i } } \pmb { \Phi } _ { j } ^ { 1 } \right) + \sum _ { t = 1 } ^ { T _ { j } } \ln p \left( y _ { j , t } | y _ { j , < t } , \pmb { \theta } _ { j } ^ { 1 ; L } \right) \right] - \sum _ { j = 1 } ^ { J } \sum _ { l = 1 } ^ { L } \mathbb { E } _ { Q } \left[ \ln \frac { q \left( \pmb { \theta } _ { j } ^ { l } | \pmb { d } _ { \pmb { \theta } _ { j } ^ { l } } \right) } { p \left( \pmb { \theta } _ { j } ^ { l } | \mathbf { e } _ { j } ^ { l } , \tau _ { 0 } \right) } \right] , } \end{array}
+$$
+
+which unites both the terms that are primarily responsible for training the recurrent hierarchical topic model component, and terms for training the neural language model component. Similar to Zhang et al. (2018), we define $q ( \pmb { \theta } _ { j } ^ { l } \ \vert \ \pmb { d } _ { \leq j } ) = \bar { \mathrm { W e i b u l l } } ( \pmb { k } _ { j } ^ { l } , \pmb { \lambda } _ { j } ^ { l } )$ , a random sample from which can be obtained by transforming standard uniform variables $\epsilon _ { j } ^ { l }$ as
+
+$$
+\pmb { \theta } _ { j } ^ { l } = \lambda _ { j } ^ { l } \big ( - \ln ( 1 - \epsilon _ { j } ^ { l } ) \big ) ^ { 1 / { k _ { j } ^ { l } } } .
+$$
+
+To capture the temporal dependencies between the topic weight vectors, both $k _ { j } ^ { l }$ and $\lambda _ { j } ^ { l }$ , from the bottom to top layers, can be expressed as
+
+$$
+\begin{array} { r } { \pmb { h } _ { j } ^ { s , l } = \mathrm { R N N } _ { \mathrm { s e n t } } ^ { l } \big ( \pmb { h } _ { j - 1 } ^ { s , l } , \pmb { h } _ { j } ^ { s , l - 1 } \big ) , \pmb { k } _ { j } ^ { l } = f _ { k } ^ { l } \big ( \pmb { h } _ { j } ^ { s , l } \big ) , \pmb { \lambda } _ { j } ^ { l } = f _ { \lambda } ^ { l } \big ( \pmb { h } _ { j } ^ { s , l } \big ) , } \end{array}
+$$
+
+where $h _ { j } ^ { s , 0 } = d _ { j } , h _ { 0 } ^ { s , l } = 0 , $ denotes the sentence-level recurrent encoder at layer $\it l$ implemented with a basic RNN cell, capturing the sequential relationship between sentences within a document, $h _ { j } ^ { s , l }$ denotes the hidden state of $\mathrm { R N N } _ { \mathrm { s e n t } } ^ { l }$ , and superscript $s$ in $h _ { j } ^ { s , l }$ denotes “sentencelevel RNN” used to distinguish the hidden state of language model in (3) . Note both $f _ { k } ^ { l }$ and $f _ { \lambda } ^ { l }$ are nonlinear functions mapping state $h _ { j } ^ { s , l }$ to the parameters of $\theta _ { j } ^ { l }$ , implemented with $f ( { \pmb x } ) =$ $\ln ( 1 + \exp ( \mathbf { W } x + b ) )$ .
+
+Rather than finding a point estimate of the global parameters $\{ \Phi ^ { l } , \Pi ^ { l } \} _ { l = 1 } ^ { L }$ of the rGBN, we adopt a hybrid inference algorithm by combining TLASGR-MCMC described in Cong et al. (2017a); Zhang et al. (2018) and our proposed recurrent variational inference network. In other words, the global parameters $\{ \bar { \boldsymbol { \Phi } } ^ { l } , \boldsymbol { \Pi } ^ { l } \} _ { l = 1 } ^ { \bar { L } }$ can be sampled with TLASGR-MCMC, while the parameters of the language model and variational recurrent inference network, denoted by $\pmb { \Omega }$ , can be updated via stochastic gradient descent (SGD) by maximizing the ELBO in (6). We describe a hybrid variational/sampling inference for rGBN-RNN in Algorithm 1 and provide more details about sampling $\{ \Phi ^ { l } , \pmb { \Pi } ^ { l } \} _ { l = 1 } ^ { L }$ with TLASGR-MCMC in Appendix C. We defer the details on model complexity to Appendix E.
+
+To sum up, as shown in Fig. 1(c), the proposed rGBN-RNN works with a recurrent variational autoencoder inference framework, which takes the document context of the $j$ th sentence within a document as input and learns hierarchical topic weight vectors $\theta _ { j } ^ { 1 : L }$ that evolve sequentially with $j$ The learned topic vectors in different layer are then used to reconstruct the document context input and as an additional feature for the language model to generate the $j$ th sentence.
+
+# 3 EXPERIMENTAL RESULTS
+
+We consider three publicly available corpora, including APNEWS, IMDB, and BNC. The links, preprocessing steps, and summary statistics for them are deferred to Appendix D. We consider a recurrent variational inference network for rGBN-RNN to infer $\theta _ { j } ^ { l }$ , as shown in Fig. 1(c), whose number of hidden units in (8) are set the same as the number of topics in the corresponding layer.
+
+Following Lau et al. (2017), word embeddings are pre-trained 300-dimension word2vec Google News vectors (https://code.google.com/archive/p/word2vec/). Dropout with a rate of 0.4 is used to the input of the stacked-RNN at each layer, i.e., $\dot { a } _ { j , t } ^ { l }$ or $W _ { e } \left[ \boldsymbol { x } _ { j , t } \right]$ in (3). The gradients are clipped if the norm of the parameter vector exceeds 5. We use the Adam optimizer (Kingma & Ba, 2015) with learning rate $1 0 ^ { - 3 }$ . The length of an input sentence is fixed to 30. We set the mini-batch size as 8, number of training epochs as 5, and scaling hyperparameter $\tau _ { 0 }$ as 1. Code in TensorFlow is provided.
+
+# 3.1 QUANTITATIVE COMPARISON
+
+Perplexity: For fair comparison, we use standard language model perplexity as the evaluation metric, by considering the following baselines: $( i )$ a standard LSTM language model (Hochreiter & Schmidhuber, 1997); (ii) LCLM (Tian & Cho, 2016), a larger-context language model that incorporates context from preceding sentences, which are treated as a bag of words; (iii) a standard LSTM language model incorporating the topic information of a separately trained LDA (LDA+LSTM); (iv) Topic-RNN (Dieng et al., 2017), a hybrid model rescoring the prediction of the next word by incorporating the topic information through a linear transformation; $( \nu )$ TDLM (Lau et al., 2017), a joint learning framework which learns a convolutional based topic model and a language model simultaneously. (vi) TCNLM (Wang et al., 2018), which extracts the global semantic coherence of a document via a neural topic model, with the probability of each learned latent topic further adopted to build a mixture-of-experts language model; (vii) TGVAE (Wang et al., 2019), combining a variational auto-encoder based neural sequence model with a neural topic model; (viii) GBN-RNN, a simplified rGBN-RNN that removes the recurrent structure of its rGBN component.
+
+For rGBN-RNN, to ensure the information about the words in the $j$ th sentence to be predicted is not leaking through the sequential document context vectors at the testing stage, the input $d _ { j }$ in (8) only summarizes the preceding sentences $S _ { < j }$ . For GBN-RNN, following TDLM (Lau et al., 2017) and TCNLM (Wang et al., 2018), all the sentences in a document, excluding the one being predicted, are used to obtain the BoW document context. As shown in Table 1, rGBN-RNN outperforms all baselines, and the trend of improvement continues as its number of layers increases, indicating the effectiveness of assimilating recurrent hierarchical topic information. rGBN-RNN consistently outperforms GBN-RNN, suggesting the benefits of exploiting the sequential dependencies of the sentence-contexts for language modeling. Moreover, comparing Table 1 and Table 4 of Appendix E suggests rGBN-RNN, with its hierarchical and temporal topical guidance, achieves better performance with fewer parameters than comparable RNN-based baselines.
+
+Note that for language modeling, there has been significant recent interest in replacing RNNs with Transformer (Vaswani et al., 2017), which consists of stacked multi-head self-attention modules, and its variants (Dai et al., 2019; Devlin et al., 2019; Radford et al., 2018; 2019). While Transformer based language models have been shown to be powerful in various natural language processing tasks, they often have significantly more parameters, require much more training data, and take much longer to train than RNN-based language models. For example, Transformer-XL with 12L and that with 24L (Dai et al., 2019), which improve Transformer to capture longer-range dependencies, have 41M and 277M parameters, respectively, while the proposed rGBN-RNN with three stochastic hidden layers has as few as $\cdot$ parameters, as shown in Table 4, when used for language modeling. From a structural point-of-view, we consider the proposed rGBN-RNN as complementary to rather than competing with Transformer based language models, and consider replacing RNN with Transformer to construct rGBN guided Transformer as a promising future extension.
+
+BLEU: Following Wang et al. (2019), we use test-BLEU to evaluate the quality of generated sentences with a set of real test sentences as the reference, and self-BLEU to evaluate the diversity of the generated sentences (Zhu et al., 2018). Given the global parameters of the deep recurrent topic model (rGBN) and language model, we can generate the sentences by following the data generation process of rGBN-RNN: we first generate topic weight vectors $\theta _ { j } ^ { L }$ randomly and then downward propagate it through the rGBN as in (2) to generate $\theta _ { j } ^ { < L }$ . By assimilating the random draw topic weight vectors with the hidden states of the language model in each layer depicted in (3), we generate a corresponding sentence, where we start from a zero hidden state at each layer in the language model, and sample words sequentially until the end-of-the-sentence symbol is generated. Comparisons of the BLEU scores between different methods are shown in Fig. 2, using the benchmark tool in Texygen (Zhu et al., 2018); We show below BLEU-3 and BLEU-4 for BNC and defer the analogous plots for IMDB and APNEWS to Appendix G and H. Note we set the validation dataset as the ground-truth.
+
+Table 1: Comparison of perplexity on three different datasets.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">LSTM Size</td><td rowspan="2">Topic Size</td><td colspan="3">Perplexity</td></tr><tr><td>APNEWS</td><td>IMDB</td><td>BNC</td></tr><tr><td rowspan="2">LCLM(Tian &amp; Cho,2016)</td><td>600</td><td></td><td>54.18</td><td>67.78</td><td>96.50</td></tr><tr><td>900-900</td><td>一</td><td>50.63</td><td>67.86</td><td>87.77</td></tr><tr><td rowspan="2">LDA+LSTM (Lau et al., 2017)</td><td>600</td><td>100</td><td>55.52</td><td>69.64</td><td>96.50</td></tr><tr><td>900-900</td><td>100</td><td>50.75</td><td>63.04</td><td>87.77</td></tr><tr><td rowspan="2">TopicRNN (Dieng et al.,2017)</td><td>600</td><td>100</td><td>54.54</td><td>67.83</td><td>93.57</td></tr><tr><td>900-900</td><td>100</td><td>50.24</td><td>61.59</td><td>84.62</td></tr><tr><td rowspan="2">TDLM (Lau et al., 2017)</td><td>600</td><td>100</td><td>52.75</td><td>63.45</td><td>85.99</td></tr><tr><td>900-900</td><td>100</td><td>48.97</td><td>59.04</td><td>81.83</td></tr><tr><td>TCNLM (Wang et al.,2018)</td><td>600</td><td>100</td><td>52.63</td><td>62.64</td><td>86.44</td></tr><tr><td rowspan="2">TGVAE (Wang et al.,2019)</td><td>900-900</td><td>100</td><td>47.81</td><td>56.38</td><td>80.14</td></tr><tr><td>600</td><td>50</td><td>48.73</td><td>57.11</td><td>87.86</td></tr><tr><td rowspan="2">basic-LSTM(Hochreiter &amp; Schmidhuber,1997)</td><td>600</td><td>一</td><td>64.13</td><td>72.14</td><td>102.89</td></tr><tr><td>900-900</td><td></td><td>58.89</td><td>66.47</td><td>94.23</td></tr><tr><td rowspan="2">GBN-RNN</td><td>900-900-900</td><td>一</td><td>60.13</td><td>65.16</td><td>95.73</td></tr><tr><td>600 600-512</td><td>100</td><td>47.42</td><td>57.01</td><td>86.39</td></tr><tr><td rowspan="3"></td><td></td><td>100-80</td><td>44.64</td><td>55.42</td><td>82.95</td></tr><tr><td>600-512-256</td><td>100-80-50</td><td>44.35</td><td>54.53</td><td>80.25</td></tr><tr><td>600</td><td>100</td><td>46.35</td><td>55.76</td><td>81.94</td></tr><tr><td rowspan="2">rGBN-RNN</td><td>600-512</td><td>100-80</td><td>43.26</td><td>53.82</td><td>80.25</td></tr><tr><td>600-512-256</td><td>100-80-50</td><td>42.71</td><td>51.36</td><td>79.13</td></tr></table>
+
+![](images/689a2217afebcff19b79da4e367570d7c586116abefb3572d628c5ba49b631f8.jpg)
+
+![](images/14d6b67e628492ad46630216b1691a87d38768d4c64bd9fb949aadeb6911e8c4.jpg)  
+Figure 2: BLEU scores of different methods for BNC. $\mathbf { X }$ -axis denotes test-BLEU, y-axis self-BLEU, and a better BLEU score would fall within the lower right corner.
+
+![](images/c18ac3637092c8dc71298ca34bffdcf08ffbd1c277ffbd215b8990ecc475f69d.jpg)  
+Figure 3: Visualization of the $L _ { 2 }$ -norm of the hidden states of the language model of rGBN-RNN, shown in the top-row, and that of GBN-RNN, shown in the bottom row.
+
+For all datasets, it is clear that rGBN-RNN yields both higher test-BLEU and lower self-BLEU scores than related methods do, indicating the stacked-RNN based language model in rGBN-RNN generalizes well and does not suffer from mode collapse (i.e., low diversity).
+
+# 3.2 QUALITATIVE ANALYSIS
+
+Hierarchical structure of language model: In Fig. 3, we visualize the hierarchical multi-scale structures learned with the language model of rGBN-RNN and that of GBN-RNN, by visualizing the $L _ { 2 }$ -norm of the hidden states in each layer, while reading a sentence from the APNEWS validation set as “the service employee international union asked why cme group needs tax relief when it is making huge amounts of money?” As shown in Fig. 3(a), in the bottom hidden layer (h1), the $\cdot$ norm sequence varies quickly from word to word, except within short phrases such “service employee”, “international union,” and “tax relief,” suggesting layer h1 is in charge of capturing short-term local dependencies. By contrast, in the top hidden layer (h3), the $L _ { 2 }$ norm sequence varies slowly and exhibits semantic/syntactic meaningful long segments, such as “service employee international union,” “asked why cme group needs tax relief,” “when it is,” and “making huge amounts of,” suggesting that layer h3 is in charge of capturing long-range dependencies. Therefore, the language model in rGBN-RNN can allow more specific information to transmit through lower layers, while allowing more general higher level information to transmit through higher layers. Our proposed model have the ability to learn hierarchical structure of the sequence, despite without designing the multiscale RNNs on purpose like Chung et al. (2017). We also visualize the language model of GBN-RNN in Fig. 3(b); with much less smoothly time-evolved deeper layers, GBN-RNN fails to utilize its stacked RNN structure as effectively as rGBN-RNN does. This suggests that the language model is much better trained in rGBN-RNN than in GBN-RNN for capturing long-range temporal dependencies, which helps explain why rGBN-RNN exhibits clearly boosted BLEU scores in comparison to GBN-RNN.
+
+![](images/c6c87135119fc368e16b91c16511c2df95d97a5e3e106d36f41782b51347d109.jpg)  
+Figure 4: Topics and their temporal trajectories inferred by a three-hidden-layer rGBN-RNN from the APNEWS dataset, and the generated sentences under topic guidance (best viewed in color). Top words of each topic at layer 3, 2, 1 are shown in orange, yellow and blue boxes respectively, and each sentence is shown in a dotted line box labeled with the corresponding topic index. Sentences generated with a combination of topics in different layers are at the bottom of the figure.
+
+Hierarchical topics: We present an example topic hierarchy inferred by a three-layer rGBN-RNN from APNEWS. In Fig. 4, we select a large-weighted topic at the top hidden layer and move down the network to include any lower-layer topics connected to their ancestors with sufficiently large weights. Horizontal arrows link temporally related topics at the same layer, while top-down arrows link hierarchically related topics across layers. For example, topic 48 of layer 3 on “budget, lawmakers, gov., revenue,” is related not only in hierarchy to topic 57 on “lawmakers, pay, proposal, legislation” and topic 60 of the lower layer on “budget, gov., revenue, vote, costs, mayor,” but also in time to topic 35 of the same layer on “democratic, taxes, proposed, future, state.” Highly interpretable hierarchical relationships between the topics at different layers, and temporal relationships between the topics at the same layer are captured by rGBN-RNN, and the topics are often quite specific semantically at the bottom layer while becoming increasingly more general when moving upwards.
+
+Sentence generation under topic guidance: Given the learned rGBN-RNN, we can sample the sentences both conditioning on a single topic of a certain layer and on a combination of the topics from different layers. Shown in the dotted-line boxes in Fig. 4, most of the generated sentences conditioned on a single topic or a combination of topics are highly related to the given topics in terms of their semantical meanings but not necessarily in key words, indicating the language model is successfully guided by the recurrent hierarchical topics. These observations suggest that rGBN-RNN has successfully captured syntax and global semantics simultaneously for natural language generation.
+
+Sentence/paragraph generation conditioning on a paragraph: Given the GBN-RNN and rGBNRNN learned on APNEWS, we further present the generated sentences conditioning on a paragraph, as shown in Fig. 5. To randomly generate sentences, we encode the paragraph into a hierarchical latent representation and then feed it into the stacked-RNN. Besides, we can generate a paragraph with rGBN-RNN, using its recurrent inference network to encode the paragraph into a dynamic hierarchical latent representation, which is fed into the language model to predict the word sequence
+
+# Document
+
+。 the senate sponsor (…) , a house committee last week removed photo ids issued by public colleges and universities from the measure sponsored by republican rep. susan lynn , who said she agreed with the change . the house approved the bill on a 65-30 vote on monday evening . but republican sen. bill ketron in a statement noted that the upper chamber overwhelmingly rejected efforts to take student ids out of the bill when it passed 21-8 earlier this month . ketron said he would take the bill to conference committee if needed .
+
+# Generated Sentences with GBN-RNN
+
+$^ { \circ }$ if the house and senate agree , it will be the first time they 'll have to seek their first meeting .   
+。 the proposal would also give lawmakers with more money to protect public safety , he said .
+
+# Generated Sentences with rGBN-RNN
+
+the proposal , which was introduced in the house on a vote on wednesday , has already passed the senate floor to the house the city commission voted last week to approve the law , which would have allowed the council to approve the new bill .
+
+# Generated temporal Sentences with rGBN-RNN (Paragraph)
+
+senate president pro tem joe scarnati said the governor 's office has never resolved the deadline for a vote in the house . the proposal is a new measure version of the bill to enact a senate committee to approve the emergency manager ‘s emergency license . the house gave the bill to six weeks of testimony , but the vote now goes to the full house for consideration . jackson signed his paperwork wednesday with the legislature .the proposal would also give lawmakers with more money to protect public safety , he said . "a spokesman for the federal department of public safety says it has been selected for a special meeting for the state senate to investigate his proposed law . a new state house committee has voted to approve a measure to let idaho join a national plan to ban private school systems at public schools . the campaign also launched a website at the university of california , irvine , which are studying the current proposal .
+
+Figure 5: Examples of generated sentences and paragraph conditioned on a document from APNEWS (green denotes novel words, blue the key words in document and generated sentences.)
+
+in each sentence of the input paragraph. It is clear that both the proposed GBN-RNN and rGBN-RNN can successfully capture the key textual information of the input paragraph, and generate diverse realistic sentences. Interestingly, the rGBN-RNN can generate semantically coherent paragraphs, incorporating contextual information both within and beyond the sentences. Note that with the topics that extract the document-level word cooccurrence patterns, our proposed models can generate semantically-meaningful words, which may not exist in the original document.
+
+# 4 CONCLUSION
+
+We propose a recurrent gamma belief network (rGBN) guided neural language modeling framework, a novel method to learn a language model and a deep recurrent topic model simultaneously. For scalable inference, we develop hybrid SG-MCMC and recurrent autoencoding variational inference, allowing efficient end-to-end training. Experiments results conducted on real world corpora demonstrate that the proposed models outperform a variety of shallow-topic-model-guided neural language models, and effectively generate the sentences from the designated multi-level topics or noise, while inferring interpretable hierarchical latent topic structure of document and hierarchical multiscale structures of sequences. For future work, we plan to extend the proposed models to specific natural language processing tasks, such as machine translation, image paragraph captioning, and text summarization. Another promising extension is to replace the stacked-RNN in rGBN-RNN with Transformer, i.e., constructing an rGBN guided Transformer as a new larger-context neural language model.
+
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+
+# A THE GBN-RNN
+
+GBN-RNN: $\{ y _ { 1 : T } , d \}$ denotes a sentence-context pair, where $d \in \mathbb { Z } _ { + } ^ { V _ { c } }$ represents the document-level context as a word frequency count vector, the vth element of which counts the number of times the vth word in the vocabulary appears in the document excluding sentence $y _ { 1 : T }$ . The hierarchical model of a $L$ -hidden-layer GBN, from top to bottom, is expressed as
+
+$$
+\begin{array} { r l } & { \pmb { \theta } ^ { L } \sim \mathrm { G a m } \left( r , c ^ { L + 1 } \right) , \ldots , \pmb { \theta } ^ { l } \sim \mathrm { G a m } \left( \Phi ^ { l + 1 } \pmb { \theta } ^ { l + 1 } , c ^ { l + 1 } \right) , \ldots , } \\ & { \pmb { \theta } ^ { 1 } \sim \mathrm { G a m } \left( \Phi ^ { 2 } \pmb { \theta } ^ { 2 } , c ^ { 2 } \right) , \ d \sim \mathrm { P o i s } \left( \Phi ^ { 1 } \pmb { \theta } ^ { 1 } \right) , } \end{array}
+$$
+
+The stacked-RNN based language model described in (3) is also used in GBN-RNN.
+
+Statistical inference: To infer GBN-RNN, we consider a hybrid of stochastic gradient MCMC (Welling & Teh, 2011; Patterson & Teh, 2013; Li et al., 2015; Ma et al., 2015; Cong et al., 2017a), used for the GBN topics $\phi _ { k } ^ { l }$ , and auto-encoding variational inference (Kingma & Welling, 2013; Rezende et al., 2014), used for the parameters of both the inference network (encoder) and RNN. More specifically, GBN-RNN generalizes Weibull hybrid auto-encoding inference (WHAI) of Zhang et al. (2018): it uses a deterministic-downward-stochastic-upward inference network to encode the bag-of-words representation of $^ d$ into the latent topic-weight variables $\pmb { \theta } ^ { l }$ across all hidden layers, which are fed into not only GBN to reconstruct $\textbf { \em d }$ , but also a stacked RNN in language model, as shown in (3), to predict the word sequence in $y _ { 1 : T }$ . The topics $\phi _ { k } ^ { l }$ can be sampled with topic-layeradaptive stochastic gradient Riemannian (TLASGR) MCMC, whose details can be found in Cong et al. (2017a); Zhang et al. (2018), omitted here for brevity. Given the sampled topics $\phi _ { k } ^ { l }$ , the joint marginal likelihood of $\{ y _ { 1 : T } , d \}$ is defined as
+
+$$
+p \left( \boldsymbol { y } _ { 1 : T } , d \mid \{ \Phi ^ { l } \} _ { l } \right) = \int p \left( d \mid \Phi ^ { 1 } \theta ^ { 1 } \right) \left[ \prod _ { t = 1 } ^ { T } p \left( \boldsymbol { y } _ { t } \mid \boldsymbol { y } _ { 1 : t - 1 } , \boldsymbol { \theta } ^ { 1 : L } \right) \right] \left[ \prod _ { l = 1 } ^ { L } p \left( \boldsymbol { \theta } ^ { l } \mid \Phi ^ { l + 1 } \boldsymbol { \theta } ^ { l + 1 } \right) \right] d \boldsymbol { \theta } ^ { 1 : L } .
+$$
+
+For efficient inference, an inference network as $\begin{array} { r } { Q = \prod _ { l = 1 } ^ { L } q ( \pmb { \theta } ^ { l } | \pmb { d } , \pmb { \Phi } ^ { l + 1 } \pmb { \theta } ^ { l + 1 } ) } \end{array}$ is used to provide a ELBO of the log joint marginal likelihood as
+
+$$
+{ \overset { , } { \left( } { y _ { 1 : T } , d \right) } } = \mathbb { E } _ { Q } \left[ \ln p \left( d \mid \Phi ^ { 1 } \theta ^ { 1 } \right) + \sum _ { t = 1 } ^ { T } \ln p \left( y _ { t } \mid y _ { 1 : t - 1 } , \theta ^ { 1 : L } \right) \right] - \sum _ { l = 1 } ^ { L } \mathbb { E } _ { Q } \left[ \ln { \frac { q \left( \theta ^ { l } \mid d , \Phi ^ { l + 1 } \theta ^ { l + 1 } \right) } { p \left( \theta ^ { l } \mid \Phi ^ { l + 1 } \theta ^ { l + 1 } \right) } } \right]
+$$
+
+and the training is performed by maximizing $\mathbb { E } _ { p _ { \mathrm { d a t a } } ( y _ { 1 : T } , d ) } [ L ( y _ { 1 : T } , d ) ]$ ; following Zhang et al. (2018), we define $q ( \pmb { \theta } ^ { l } \mid d , \pmb { \Phi } ^ { l + 1 } , \pmb { \theta } ^ { l + 1 } ) = \mathrm { W e i b u l l } ( \pmb { k } ^ { l } + \pmb { \Phi } ^ { l + 1 } \pmb { \theta } ^ { l + 1 } , \pmb { \lambda } ^ { l } )$ , where both $k ^ { l }$ and $\lambda ^ { l }$ are deterministically transformed from $^ d$ using neural networks. Distinct from a usual variational auto-encoder whose inference network has a pure bottom-up structure, the inference network here has a determisticupward–stoachstic-downward ladder structure (Zhang et al., 2018).
+
+# B THE COUPLING VECTOR
+
+Following Lau et al. (2017), the $\mathbf { \Delta } _ { \cdot \cdot } ^ { a _ { j , t } ^ { l } } = g ^ { l } \left( h _ { j , t } ^ { l } , \theta _ { j } ^ { l } \right)$ can be implemented with a gating unit similar to a GRU (Cho et al., 2014), describe as
+
+$$
+\begin{array} { r l } & { z _ { j , t } ^ { l } = \sigma \left( \mathbf { W } _ { z } ^ { l } \pmb { \theta } _ { j } ^ { l } + \mathbf { U } _ { z } ^ { l } \pmb { h } _ { j , t } ^ { l } + \mathbf { b } _ { z } ^ { l } \right) } \\ & { r _ { j , t } ^ { l } = \sigma \left( \mathbf { W } _ { r } ^ { l } \pmb { \theta } _ { j } ^ { l } + \mathbf { U } _ { r } ^ { l } \pmb { h } _ { j , t } ^ { l } + \mathbf { b } _ { r } ^ { l } \right) } \\ & { \hat { h } _ { j , t } ^ { l } = \operatorname { t a n h } \left( \mathbf { W } _ { h } ^ { l } \pmb { \theta } _ { j } ^ { l } + \mathbf { U } _ { h } ^ { l } \left( r _ { j , t } ^ { l } \odot { h } _ { j , t } ^ { l } \right) + \mathbf { b } _ { h } ^ { l } \right) } \\ & { a _ { j , t } ^ { l } = \left( 1 - z _ { j , t } ^ { l } \right) \odot { h } _ { j , t } ^ { l } + z _ { j , t } ^ { l } \odot \hat { h } _ { j , t } ^ { l } } \end{array}
+$$
+
+# C SGMCMC FOR RGBN-RNN
+
+To allow for scalable inference, we apply the topic-layer-adaptive stochastic gradient Riemannian (TLASGR) MCMC algorithm described in Cong et al. (2017a); Zhang et al. (2018), which can be used to sample simplex-constrained global parameters Cong et al. (2017b) in a mini-batch based manner. It improves its sampling efficiency via the use of the Fisher information matrix (FIM)
+
+Girolami & Calderhead (2011), with adaptive step-sizes for the latent factors and transition matrices of different layers. In this section, we discuss how to update the global parameters $\{ \Phi ^ { l } , \Pi ^ { l } \} _ { l = 1 } ^ { L }$ of rGBN in detail and give a complete one in Algorithm in 1.
+
+Sample the auxiliary counts: This step is about the “backward” and “upward” pass. Let us denote $\_$ Kl , $\cdot$ , and $x _ { k j } ^ { ( 1 , 1 ) } = d _ { v j }$ , where $-$ is shown in (2). Working backward for $\cdot$ and upward for $\cdot$ , we draw
+
+$$
+\begin{array} { r l } & { ( A _ { k 1 j } ^ { l } , . . . , . , A _ { k K _ { l } j } ^ { l } ) \sim \mathrm { M u l t i } \left( x _ { k j } ^ { ( l , l ) } ; \frac { \phi _ { k 1 } ^ { l } \theta _ { 1 j } ^ { l } } { \sum _ { k _ { l } = 1 } ^ { K _ { l } } \phi _ { k k _ { l } } ^ { l } \theta _ { k l } ^ { l } } , . . . , \frac { \phi _ { k K _ { l } } ^ { l } \theta _ { K l } ^ { l } } { \sum _ { k _ { l } = 1 } ^ { K _ { l } } \phi _ { k l } ^ { l } \theta _ { k l } ^ { l } } \right) , } \\ & { x _ { k j } ^ { l + 1 } \sim \mathrm { C R T } \left[ A _ { \cdot k j } ^ { l } + Z _ { \cdot k , j + 1 } ^ { l } , \tau _ { 0 } \left( \sum _ { k _ { l + 1 } = 1 } ^ { K _ { l + 1 } } \phi _ { k k _ { l + 1 } } ^ { l + 1 } \theta _ { k _ { l + 1 } j } ^ { l + 1 } + \sum _ { k _ { l } = 1 } ^ { K _ { l } } \pi _ { k k _ { 1 } } ^ { l } \theta _ { k _ { 1 } , j - 1 } ^ { l } \right) \right] . } \end{array}
+$$
+
+Note that via the deep structure, the latent counts xl+1kj will be influenced by the effects from both time $\cdot$ at layer $\cdot$ and time $j$ at layer ${ \it l } + 1$ . With $\_$ and $p _ { 2 } : =$ $\cdot$ , we can sample the latent counts at layer $\cdot$ and ${ \it l } + 1$ by
+
+$$
+( x _ { k j } ^ { l + 1 , l } , x _ { k j } ^ { l + 1 , l + 1 } ) \sim \mathrm { M u l t i } \left( x _ { k j } ^ { l + 1 } , p _ { 1 } / ( p _ { 1 } + p _ { 2 } ) , p _ { 2 } / ( p _ { 1 } + p _ { 2 } ) \right) ,
+$$
+
+and then draw
+
+$$
+-
+$$
+
+In rGBN, the prior and the likelihood of $\{ \Phi ^ { l } \} _ { l = 1 } ^ { L }$ is very similar with $\{ \boldsymbol { \Pi } ^ { l } \} _ { l = 1 } ^ { L }$ , so we also apply the TLASGR MCMC sampling algorithm on both of them conditioned on the auxiliary counts.
+
+Sample the hierarchical components $\{ \Phi ^ { l } \} _ { l = 1 } ^ { L }$ : For $\phi _ { k } ^ { l }$ , the $k$ th column of the loading matrix $\Phi ^ { l }$ of layer $l$ , its sampling can be efficiently realized as
+
+$$
+\begin{array} { r l r } {  { ( \phi _ { k } ^ { l } ) _ { n + 1 } = [ ( \phi _ { k } ^ { l } ) _ { n } + \frac { \varepsilon _ { n } } { P _ { k } ^ { l } } [ ( \rho \tilde { A } _ { : k \cdot } ^ { l } + \eta _ { 0 } ^ { l } ) - ( \rho \tilde { A } _ { : k \cdot } ^ { l } + K _ { l - 1 } \eta _ { 0 } ^ { l } ) ( \phi _ { k } ^ { l } ) _ { n } ] } } \\ & { } & { + \ N ( 0 , \frac { 2 \varepsilon _ { n } } { P _ { k } ^ { l } } [ \mathrm { d i a g } ( \phi _ { k } ^ { l } ) _ { n } - ( \phi _ { k } ^ { l } ) _ { n } ( \phi _ { k } ^ { l } ) _ { n } ^ { T } ] ) ] _ { \angle } , } \end{array}
+$$
+
+where $P _ { k } ^ { l }$ is calculated using the estimated FIM, $\begin{array} { r } { \tilde { A } _ { k _ { l } j . } ^ { l } = \sum _ { j = 1 } ^ { J } A _ { k _ { l } k { j } } ^ { l } , \tilde { A } _ { : k \cdot } ^ { l } = \{ \tilde { A } _ { 1 j \cdot } ^ { l } , \cdot \cdot \cdot , \tilde { A } _ { K _ { l } j \cdot } ^ { l } \} ^ { T } } \end{array}$ and $\begin{array} { r } { \tilde { A } _ { \cdot k \cdot } ^ { l } = \sum _ { k _ { l } = 1 } ^ { k _ { l } } \tilde { A } _ { k _ { l } j } ^ { l } . } \end{array}$ , $A _ { k _ { l } k j } ^ { l }$ l  l comes from the augmented latent counts $A ^ { l }$ in (13), $\eta _ { 0 } ^ { l }$ l denote the prior of $\phi _ { k } ^ { l }$ , and $[ \cdot ] _ { \angle }$ denotes a simplex constraint.
+
+Sample the transmission matrix $\{ \boldsymbol { \Pi } ^ { l } \} _ { l = 1 } ^ { L }$ : For $\pi _ { k } ^ { l }$ , the $k$ th column of the transition matrix $\Pi ^ { l }$ of layer $l$ , its sampling can be efficiently realized as
+
+$$
+\begin{array} { r l } & { \left( \pi _ { k } ^ { l } \right) _ { n + 1 } = \biggr [ \left( \pi _ { k } ^ { l } \right) _ { n } + \frac { \varepsilon _ { n } } { M _ { k } ^ { l } } \left[ \left( \rho \tilde { Z } _ { : k \cdot } ^ { l } + \eta _ { : k } ^ { l } \right) - \left( \rho \tilde { Z } _ { : k \cdot } ^ { l } + \eta _ { . k } ^ { l } \right) \left( \pi _ { k } ^ { l } \right) _ { n } \right] } \\ & { \qquad + \mathcal { N } \left( 0 , \frac { 2 \varepsilon _ { n } } { M _ { k } ^ { l } } \left[ \mathrm { d i a g } ( \pi _ { k } ^ { l } ) _ { n } - ( \pi _ { k } ^ { l } ) _ { n } ( \pi _ { k } ^ { l } ) _ { n } ^ { T } \right] \right) \biggr ] _ { \angle } , } \end{array}
+$$
+
+where $M _ { k } ^ { l }$ is calculated using the estimated FIM, $\begin{array} { r } { \tilde { Z } _ { k _ { l } j . } ^ { l } = \sum _ { j = 1 } ^ { J } Z _ { k _ { l } k _ { j } } ^ { l } , \tilde { Z } _ { : k \cdot } ^ { l } = \{ \tilde { Z } _ { 1 j \cdot } ^ { l } , \cdot \cdot \cdot , \tilde { Z } _ { K _ { l } j \cdot } ^ { l } \} ^ { T } } \end{array}$ and $\begin{array} { r } { \tilde { Z } _ { \cdot k \cdot } ^ { l } = \sum _ { k _ { l } = 1 } ^ { k _ { l } } \tilde { Z } _ { k _ { l } j } ^ { l } } \end{array}$ · , $Z _ { k _ { l } k j } ^ { l }$ comes from the augmented latent counts $Z ^ { l }$ in (16), and $[ . ] _ { \angle }$ denotes a simplex constraint, and $\eta _ { ; k } ^ { l }$ denotes the prior of $\pi _ { k } ^ { l }$ , more details about TLASGR-MCMC for our proposed model can be found in Cong et al. (2017a).
+
+# D DATASETS
+
+We consider three publicly available corpora1. APNEWS is a collection of Associated Press news articles from 2009 to 2016, IMDB is a set of movie reviews collected by Maas et al. (2011), and BNC is the written portion of the British National Corpus (Consortium, 2007). Following the preprocessing steps in Lau et al. (2017), we tokenize words and sentences using Stanford CoreNLP (Klein & Manning, 2003), lowercase all word tokens, and filter out word tokens that occur less than 10 times. For the topic model, we additionally exclude stopwords2 and the top $0 . 1 \%$ most frequent words. All these corpora are partitioned into training, validation, and testing sets, whose summary statistics are provided in Table 2 of the Appendix.
+
+Table 2: Summary statistics for the datasets.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">Vocubalry</td><td colspan="3">Training</td><td colspan="3">Validation</td><td colspan="3">Testing</td></tr><tr><td>LM</td><td>TM</td><td>Docs</td><td>Sents</td><td>Tokens</td><td>Docs</td><td>Sents</td><td>Tokens</td><td>Docs</td><td>Sents</td><td>Tokens</td></tr><tr><td>APNEWS</td><td>34231</td><td>32169</td><td>50K</td><td>0.8M</td><td>15M</td><td>2K</td><td>33K</td><td>0.6M</td><td>2K</td><td>32K</td><td>0.6M</td></tr><tr><td>IMDB</td><td>36009</td><td>34925</td><td>75K</td><td>1.1M</td><td>20M</td><td>12.5K</td><td>0.18M</td><td>0.3M</td><td>12.5K</td><td>0.18M</td><td>0.3M</td></tr><tr><td>BNC</td><td>43703</td><td>41552</td><td>15K</td><td>1M</td><td>18M</td><td>1K</td><td>57K</td><td>1M</td><td>1K</td><td>66K</td><td>1M</td></tr></table>
+
+# E COMPLEXITY OF RGBN-RNN
+
+The proposed rGBN-RNN consists of both language model and topic model compontopic model component, there are the global parameters of rGBN (decoder), including $\{ \Phi ^ { l } , \Pi ^ { l } \} _ { l = 1 } ^ { L }$ in (2) , and the parameters of the variational recurrent inference network (encoder), consisting of $\cdot$ , $f _ { k } ^ { l }$ , and $\cdot$ in (8). The language model component is parameterized by $\mathrm { L S T M } _ { \mathrm { w o r d } } ^ { l }$ in (3) and the coupling vectors $\cdot$ described in Appendix B. We summarize in Table 3 the complexity of rGBN-RNN (ignoring all bias terms), where $V$ denotes the vocabulary size of the language model, $\cdot$ the dimension of word embedding vectors, $V _ { c }$ the size of the vocabulary of the topic model that excludes stop words, $H _ { l } ^ { w }$ the number of hidden units of the word-level LSTM at layer $\cdot$ (stacked-RNN language model), $H _ { l } ^ { s }$ the number of hidden units of the sentence-level RNN at layer $\cdot$ (variational recurrent inference network), and $\cdot$ the number of topics at layer $\it l$ .
+
+Table 4 further compares the number of parameters between various RNN-based language models, where we follow the convention to ignore the word embedding layers. Some models in Table 1 are not included here, because we could not find sufficient information from their corresponding papers or code to accurately calculate the number of model parameters. Note when used for language generation at the testing stage, rGBN-RNN no longer needs its topics $\{ \Phi ^ { l } \}$ , whose parameters are hence not counted. Note the number of parameters of the topic model component is often dominated by that of the language model component.
+
+Table 3: Complexity of the three-layer rGBN-RNN.   
+
+<table><tr><td rowspan="2">Component Param</td><td colspan="2">Language Model</td><td colspan="2"></td><td colspan="4">Topic Model</td></tr><tr><td>LSTMword in (3)</td><td>glinB</td><td>Φin (2)</td><td>II in (2)</td><td></td><td>RNNsent in (8)</td><td>f in (8)</td><td>f in (8)</td></tr><tr><td>Layer1</td><td>O(4×(E+H) ×H&quot;)</td><td>O(3×(K1+H)×H1)</td><td>O(Vc × K1)</td><td>O(K1 × K1)</td><td></td><td>O(Vc+H𝑖) × Hi</td><td>O(Hi)</td><td>O(K1 x Hi)</td></tr><tr><td>Layer2</td><td>O(4×(H+H)×H²)</td><td>O(3 × (K2+H) × H)</td><td>O(K1 × K2)</td><td>O(K2 ×K2)</td><td>O((Hi +H²)×H)</td><td></td><td>O(H)</td><td>O(K2 ×H)</td></tr><tr><td>Layer3</td><td>O(4×(H+H)×H))</td><td>1O(3×(K3+H)×H)</td><td></td><td></td><td>[O(K2×K3)|O(K3 × K3)|O(H+H) × H)</td><td></td><td>O(H)</td><td>O(K ×H)</td></tr></table>
+
+Table 4: Comparison of the number of parameters of different models when used for language generation.   
+
+<table><tr><td>Model</td><td>LSTM Size</td><td>Topic Size</td><td># LM Param</td><td>#TMParam</td><td># All Param</td></tr><tr><td>TDLM (Lau et al., 2017)</td><td>600 900-900</td><td>100 100</td><td>3.35M 13.38M</td><td>0.019M 0.019M</td><td>3.37M 13.40M</td></tr><tr><td>basic-LSTM (Hochreiter &amp; Schmidhuber,1997)</td><td>600 900-900 900-900-900</td><td></td><td>2.16M 10.80M 17.68M</td><td></td><td>2.16M 10.80M 17.68M</td></tr><tr><td>GBN-RNN</td><td>600 600-512 600-512-256</td><td>100 100-80 100-80-50</td><td>3.40M 6.50M 7.20M</td><td>0.02M 0.04M 0.05M</td><td>3.42M 6.54M 7.25M</td></tr><tr><td>rGBN-RNN</td><td>600 600-512 600-512-256</td><td>100 100-80 100-80-50</td><td>3.40M 6.50M 7.20M</td><td>0.03M 0.06M 0.07M</td><td>3.43M 6.56M 7.27M</td></tr></table>
+
+![](images/d0f732dcd06494584cd25d57050fc337f309e1ed610f269575fe293a28ccb69d.jpg)  
+Figure 6: Topics and their temporal trajectories inferred by a three-hidden-layer rGBN-RNN from the IMDB dataset, and the generated sentences under topic guidance (best viewed in color). Top words of each topic are shown in orange, yellow and blue box, and each sentence is shown in a dotted line box labeled with the corresponding topic index. Sentences generated with a combination of topics in different layers are at the bottom of the figure.
+
+![](images/e51953d8d5fc18d93d196deb64d7eedba8a7e6e6e2a96d75c4b704e3b727ab13.jpg)  
+Figure 7: Topics and their temporal trajectories inferred by a three-hidden-layer rGBN-RNN from the BNC dataset, and the generated sentences under topic guidance (best viewed in color). Top words of each topic are shown in orange, yellow and blue box, and each sentence is shown in a dotted line box labeled with the corresponding topic index. Sentences generated with a combination of topics in different layers are at the bottom of the figure.
+
+![](images/55b975ca0a44ac5eea39ae50f5a4b8d259eecdc7e16870f52516b9082a795520.jpg)  
+Figure 8: BLEU scores of different methods for IMDB. $\mathbf { X }$ -axis denotes test-BLEU, and y-axis self-BLEU. Left panel is BLEU-3 and right is BLEU-4, and a better BLEU score would fall within the lower right corner, where black point represents mean value and circles with different colors denote the elliptical surface of probability of BLEU in a two-dimensional space.
+
+# H BLEU SCORES FOR APNEWS
+
+![](images/e8bbc3300e8cdcbc6089fe0c71b52e2f90f4cc90ee75d4b1dc906e44895a6eee.jpg)  
+Figure 9: BLEU scores of different methods for APNEWS. $\mathbf { X }$ -axis denotes test-BLEU, and y-axis self-BLEU. Left panel is BLEU-3 and right is BLEU-4, and a better BLEU score would fall within the lower right corner, where black point represents mean value and circles with different colors denote the elliptical surface of probability of BLEU in a two-dimensional space.

parse/train/HJMHpjC9Ym/HJMHpjC9Ym.md

@@ -0,0 +1,402 @@
+# BIG-LITTLE NET: AN EFFICIENT MULTI-SCALE FEATURE REPRESENTATION FOR VISUAL AND SPEECH RECOGNITION
+
+Chun-Fu (Richard) Chen, Quanfu Fan, Neil Mallinar, Tom Sercu, Rogerio Feris IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 {chenrich, qfan, neil.r.mallinar, tom.sercu1, rsferis}@us.ibm.com
+
+# ABSTRACT
+
+In this paper, we propose a novel Convolutional Neural Network (CNN) architecture for learning multi-scale feature representations with good tradeoffs between speed and accuracy. This is achieved by using a multi-branch network, which has different computational complexity at different branches with different resolutions. Through frequent merging of features from branches at distinct scales, our model obtains multi-scale features while using less computation. The proposed approach demonstrates improvement of model efficiency and performance on both object recognition and speech recognition tasks, using popular architectures including ResNet, ResNeXt and SEResNeXt. For object recognition, our approach reduces computation by $1 / 3$ while improving accuracy significantly over $1 \%$ point than the baselines, and the computational savings can be higher up to $1 / 2$ without compromising the accuracy. Our model also surpasses state-of-the-art CNN acceleration approaches by a large margin in terms of accuracy and FLOPs. On the task of speech recognition, our proposed multi-scale CNNs save $30 \%$ FLOPs with slightly better word error rates, showing good generalization across domains.
+
+# 1 INTRODUCTION
+
+Deep Convolutional Neural Network (CNN) models have achieved substantial performance gains in many computer vision and speech recognition tasks (He et al., 2016; 2017; Vinyals et al., 2017; Sercu & Goel, 2016). However, the accuracy obtained by these models usually grows proportionally with their complexity and computational cost. This poses an issue for deploying these models in applications that require real-time inferencing and low-memory footprint, such as self-driving vehicles, human-machine interaction on mobile devices, and robotics.
+
+Motivated by these applications, many methods have been proposed for model compression and acceleration, including techniques such as pruning (Dong et al., 2017; Li et al., 2017; Han et al., 2015), quantization (Hubara et al., 2016; Li & Liu, 2016), and low-rank factorization (Wen et al., 2017; Ioannou et al., 2015; Zhang et al., 2016). Most of these methods have been applied to single-scale inputs, without considering multi-resolution processing. More recently, another line of work applies dynamic routing to allocate different workloads in the networks according to image complexity (Wu et al., 2018; Wang et al., 2018; Figurnov et al., 2017; Veit & Belongie, 2018). Multi-scale feature representations have proven successful for many vision and speech recognition tasks compared to single-scale methods (Nah et al., 2017; Chen et al., 2017a; Tóth, 2017; Farabet et al., 2013); however, the computational complexity has not been addressed much in multi-scale networks.
+
+The computational cost of a CNN model has much to do with the input image size. A model, if running at half of the image size, can gain a remarkable computational saving of $7 5 \%$ . Based on this fact, we propose an efficient network architecture by combining image information at different scales through a multi-branch network. As shown in Fig. 1, our key idea is to use a high-complexity branch (accurate but costly) for low-scale feature representation and low-complexity branch (efficient but less accurate) for high-scale feature representation. The two types of features are frequently merged together to complement and enrich each other, leading to a stronger feature representation than either of them individually. We refer to the deeper branch operating at low image resolution as Big-Branch and the shallower one as Little-Branch to reflect their differences in computation. The new network architecture is thus called Big-Little Net or bL-Net for short in this paper.
+
+![](images/0574022df5e50adb5747f9dd6217fb99fafb4314cbc906cbae2227b65290d2a4.jpg)  
+Figure 1: Our proposed Big-Little Net (bL-Net) for efficient multi-scale feature representations. (a) The bL-Net stacks several Big-Little Modules. A bL-module include $K$ branches $K = 2$ in this illustration) where the $k ^ { t h }$ branch represents an image scale of $1 / 2 ^ { k }$ . ‘M’ here denotes a merging operation. (b) Our implementation of the Big-Little Module includes two branches. The Big-Branch has the same structure as the baseline model while the Little-Branch reduces the convolutional layers and feature maps by $\alpha$ and $\beta$ , respectively. Larger values of $\alpha$ and $\beta$ lead to lower computational complexity in Big-Little Net.
+
+While being structurally simple, our approach is quite effective. We demonstrate later that when bL-Net is integrated into state-of-the-art CNNs such as ResNet, ResNeXt and SEResNeXt, it yields $2 \times$ computational savings over the baselines without losing any accuracy. It also outperforms many other recently developed approaches based on more sophisticated architectures at the same FLOP count. One work that relates to ours is the Inception model (Szegedy et al., 2016a; 2017), which also leverages parallel pathways in a network building block for efficiency. However, the efficiency of these models relies on substantial use of $1 \times 1$ and separable filters (i.e. $1 \times 3$ and $3 \times 1$ ). In contrast, bL-Net is general and applicable to many architectures including Inception.
+
+The main contributions of our paper are summarized as follows:
+
+• We propose an efficient and effective multi-scale CNN architecture for object and speech recognition.   
+• We demonstrate that our approach reduces computation by $1 / 3$ in models such as ResNet and ResNeXt while improving accuracy over $1 \%$ point than the baselines, and the computational savings can be higher up to $1 / 2$ without losing any accuracy; these results outperform state-of-the-art networks that focus on CNN acceleration by a large margin at the same FLOPs.   
+We validate the proposed method on a speech recognition task, where we also achieve better word error rates while reducing the number of FLOPs by $30 \%$ .
+
+# 2 RELATED WORK
+
+Model Compression and Acceleration. Network pruning (Dong et al., 2017; Li et al., 2017) and quantization (Hubara et al., 2016; Li & Liu, 2016) are popular techniques to remove model redundancy and save computational cost. Another thread of work consists of training a sparse model directly, such as IGCv2 (Xie et al., 2018) and SCConv (Fan et al., 2017). Efficient network architectures like MobileNetV2 (Sandler et al., 2018) and ShuffleNetV2 (Ma et al., 2018) have also been explored for training compact deep networks. Other methods include knowledge distillation (Hinton et al., 2015), compression with structured matrices (Cheng et al., 2015; Sindhwani et al., 2015), and hashing (Chen et al., 2015). Dynamic routing (Wu et al., 2018; Wang et al., 2018; Figurnov et al., 2017; Veit & Belongie, 2018) has also been explored in residual networks to improve efficiency. These methods operate with single-resolution inputs, while our approach processes multi-resolution data. It could be used in tandem with these methods to further improve efficiency.
+
+Multi-Resolution Feature Representations. The notion of multi-scale feature representation can be dated back to image pyramids (Adelson et al., 1984) and scale-space theory (Lindeberg & ter Haar Romeny, 1994). More recently, several methods have proposed multi-scale CNN-based architectures for object detection and recognition. MSCNN (Cai et al., 2016), DAG-CNN (Yang & Ramanan, 2015) and FPN (Lin et al., 2017) use features at different layers to form multi-scale features. Hourglass networks (Newell et al., 2016) use a hierarchical multi-resolution model for human pose estimation. However, this approach induces a heavy workload as the complexity of each sub-network in their model is equal. Nah et al. (Nah et al., 2017) and Eigen et al. (Eigen & Fergus, 2015) combine the features from multiple networks working on different resolutions to generate multi-scale features. The overall computational cost grows along with the number of scales, leading to inefficient models. In contrast to existing methods, our approach uses different network capacities for different scales, and yields more powerful multi-scale features by fusing them at multiple levels of the network model.
+
+Closely related to our work, approaches such as (Huang et al., 2018; Saxena & Verbeek, 2016), apply multiple branches at multi-scales while aiming at reducing computational complexity. In contrast to our work, their computational gain mostly comes from early exit depending on the input image. Our approach speeds up a network constantly regardless of the input image.
+
+# 3 OUR APPROACH
+
+We develop a simple, easy-to-implement, yet very efficient and effective network architecture. It learns multi-scale feature representations by fusing multiple branches with different image scales and computational complexity. As shown in Fig. 1, we design a multi-scale feature module with the following principles: (I) each branch corresponds to a single unique image scale (or resolution); (II) the computational cost of a branch is inversely proportional to the scale. Note that the principle (II) implies that we use high-complexity networks at lower resolutions and low-complexity networks at higher resolutions for the sake of efficiency.
+
+# 3.1 BIG-LITTLE NET
+
+Big-Little Net is a sequence of Big-Little Modules, each one taking input $\mathbf { x } _ { i }$ and producing output $\mathbf { x } _ { i + 1 }$ . Within a Big-Little Module, assume we have $K$ branches working on $K$ scales $[ 1 , \bar { 1 } / 2 , 1 / 4 , . . . , 1 / 2 ^ { K - 1 } ]$ . We denote a feature map $\mathbf { x } _ { i }$ at scale $1 / 2 ^ { k }$ as $\mathbf { x } _ { i } ^ { k }$ , indicating the spatial size of $\mathbf { x } _ { i }$ downsampled by $2 ^ { k }$ with respect to the original input dimension. We use a weighted sum to combine all branches into a feature representation at scale $1 / 2 ^ { k }$ . Mathematically, the module’s output $\mathbf x _ { i + 1 }$ can be expressed by
+
+$$
+\mathbf { x } _ { i + 1 } = F \left( \sum _ { k = 0 } ^ { K - 1 } c ^ { k } S ^ { k } \left( f _ { k } \left( \mathbf { x } _ { i } ^ { k } \right) \right) \right) ,
+$$
+
+where $f _ { k } ( \cdot )$ denotes a sequence of convolutional layers. Typically for higher $k$ , $f _ { k }$ will have more convolutional layers having more feature maps. $S ^ { k } ( \cdot )$ is the operation that matches the output size of the branches, either: (1) increasing the number feature maps with a $1 \times 1$ convolution, (2) upsampling to match the output size of the $k = 0$ branch, or both. $c ^ { k }$ indicates the weighting coefficients of each scale in the merge while $F ( \cdot )$ is an optional final fusion layer like a convolutional layer.
+
+Note that branches are merged at the end of every Big-Little Module and merging the branch outputs happens at the highest resolution and highest number of feature maps between the branches. Maintaining these large intermediate states avoids information loss. A crucial aspect of this design is that through consecutive merging and downsampling, the expensive branches operating at low resolution still have access to the high resolution information, processed by the cheaper branches in the previous module.
+
+While our design is suitable for any number of networks, in this work we primarily focus on the case of two networks, i.e., $K = 2$ . We also experimented with $K > 2$ in object and speech recognition; however, the $K = 2$ case provided the best balance between accuracy and computation (See Appendix A.4 and Section 4.2 for details). Following the principles above, we propose a multinetwork architecture that integrates two branches for multi-scale feature representation. Figure 1 (b) shows an example Big-Little Net architecture.
+
+The module includes two branches, each of which represents a separate network block from a deep model (accurate but costly) and a less deep counterpart (efficient but less accurate). The two branches are fused at the end through linear combination with unit weights (i.e., $c ^ { 0 } = c ^ { 1 } = 1 . 0$ ). Before fusion, the low resolution feature maps are upsampled using bilinear interpolation to spatially match the higher-resolution counterparts $\bar { ( = { S ^ { 1 } ( \cdot ) } ) }$ . Similarly, the high resolution feature map has an additional $1 \times 1$ convolutional layer to increase the number of output channels $( = S ^ { 0 } ( \cdot ) )$ . Furthermore, since our design is based on ResNet, we add a residual block to further fuse the combined features (i.e., $F ( \cdot )$ is a residual block). For convenience, we refer to these two branches as Big-Branch (more layers and channels at low resolution) and Little-Branch (fewer layers and channels at high resolution), respectively. We also denote the module as Big-Little Module and the entire architecture as Big-Little Net or bL-Net.
+
+To control the complexity of bL-Net, we introduce two parameters to specify the complexity of the Little-Branch with respect to the Big-Branch. The Big-Branch typically follows the structure of the original network, but the Little-Branch needs to be heavily slimmed and shortened to reduce computation as it operates on high resolution. Here we use two parameters $\alpha$ and $\beta$ to control the width and depth of the Little-Branch, respectively. As shown in Fig. 1 (b), $\alpha$ specifies the reduction factor of the number of channels in the convolutional layers of Little-Branch with respect to that of the original network while $\beta$ is the reduction factor of the number of convolutional layers. Larger values of $\alpha$ and $\beta$ lead to lower complexity in bL-Net. As demonstrated later, with an appropriate choice of $\alpha$ and $\beta$ (See Table 1 and Table 3), the cost of the Little-Branch can be $1 / 6$ of the $b L$ -Net and $1 / 1 2$ of the original network while still providing sufficient complementary information to the Big-Branch.
+
+# 3.2 NETWORK MERGING
+
+We consider two options for merging the outputs of the branches. The first option is a linear combination, which joins features from two networks by addition (Newell et al., 2016). The alternative concatenates the outputs of the two networks along the channel dimension, and if needed, a $1 \times 1$ convolution can be subsequently applied to reduce the number of feature maps (Szegedy et al., 2015). Both merging approaches have their pros and cons. With linear combination, the branches can easily compensate each other, meaning each branch can activate output neurons not activated by the other. However, additional cost is added as both the size of feature maps and the number of channels in the two branches need to be adjusted to be the same before addition. On the other hand, merging by concatenation only needs to spatially align the feature maps. On the other hand, concatenation only needs to align the feature map size, however requires a $1 \times 1$ convolution reducing the number of channels after concatenation, which is a more expensive operation than the pointwise addition.
+
+While linear combination provides an immediate exchange of the activations of both branches, concatenation relies on the following layers for this exchange. This delay in exchange could possibly be problematic if the information from each branch is destructively altered before merging. For example, a nonlinearity such as ReLU would discard all activations less than zero, effectively ignoring negative features in both branches before merging. Since linear combination does not cause too much overhead and provides better accuracy, we chose linear combination as our merging approach. In Appendix A.4, we empirically show that the linear combination approach performs better than concatenation in object recognition.
+
+# 4 EXPERIMENTAL RESULTS
+
+We conducted extensive experiments, as discussed below, to validate the effectiveness of our proposed bL-Net on object and speech recognition tasks. bL-Net can be easily integrated with many modern CNNs and here we chose ResNet (He et al., 2016) as the primary architecture to evaluate our approach. For simplicity, from now on, we denote by bL-M the bL-Net using a backbone network $M$ For example, bL-ResNet-50 is the Big-Little net based on ResNet-50.
+
+# 4.1 OBJECT RECOGNITION
+
+We used the ImageNet dataset (Russakovsky et al., 2015) for all the experiments below on object recognition. This dataset is a common benchmark for object recognition, which contains 1.28 million training images and $5 0 \mathrm { k }$ validation images with labels from 1000 categories. The details of our experimental setup and the network structures for bL-ResNet-50, 101 and 152 can be found in Appendix A.1.
+
+Table 1: Complexity study of the Little-Branch $\scriptstyle { \alpha }$ and $\beta$ ) for bL-ResNet-50.   
+
+<table><tr><td>Model</td><td>Top-1 Error</td><td>FLOPs (109)</td><td>Params (106)</td></tr><tr><td>ResNet-50</td><td>23.66%</td><td>4.09</td><td>25.55</td></tr><tr><td>bL-ResNet-50 (α = 2, β = 2)</td><td>22.72%</td><td>2.91 (1.41×)</td><td>26.97</td></tr><tr><td>bL-ResNet-50(α = 2, β= 4)</td><td>22.69%</td><td>2.85 (1.44×)</td><td>26.69</td></tr><tr><td>bL-ResNet-50 (α = 4, β = 2)</td><td>23.20%</td><td>2.49 (1.64×)</td><td>26.31</td></tr><tr><td>bL-ResNet-50 (α= 4, β= 4)</td><td>23.15%</td><td>2.48 (1.65×)</td><td>26.24</td></tr></table>
+
+Table 2: Performance comparison for bL-ResNet, bL-ResNeXt and bL-SEResNeXt.   
+
+<table><tr><td>Model</td><td>Top-1 Error</td><td>FLOPs (109)</td><td>Params (106)</td><td>Speed (ms/batch)t</td></tr><tr><td>ResNet-101</td><td>21.95%</td><td>7.80</td><td>44.54</td><td>186</td></tr><tr><td>bL-ResNet-101 (α= 2,β= 4)</td><td>21.80%</td><td>3.89 (2.01×)</td><td>41.85</td><td>140 (1.33×)</td></tr><tr><td>bL-ResNet-101@256(α = 2,β= 4)</td><td>21.04%</td><td>5.08 (1.54×)</td><td>41.85</td><td>162 (1.15×)</td></tr><tr><td>ResNet-152</td><td>21.51%</td><td>11.51</td><td>60.19</td><td>266</td></tr><tr><td>bL-ResNet-152(α = 2,β =4)</td><td>21.16%</td><td>5.04 (2.28×)</td><td>57.36</td><td>178 (1.49×)</td></tr><tr><td>bL-ResNet-152@256 (α= 2, β=4)</td><td>20.34%</td><td>6.58 (1.75x)</td><td>57.36</td><td>205 (1.30×)</td></tr><tr><td>ResNeXt-50 (32×4d)</td><td>22.20%</td><td>4.23</td><td>25.03</td><td>157</td></tr><tr><td>bL-ResNeXt-50 (32×4d) (α =2,β= 4)</td><td>21.60%</td><td>3.03 (1.40×)</td><td>26.19</td><td>125 (1.26×)</td></tr><tr><td>bL-ResNeXt-50@256(α = 2, β= 4)</td><td>20.96%</td><td>3.95 (1.08×)</td><td>26.19</td><td>153 (1.03×)</td></tr><tr><td>ResNeXt-101 (32×4d)</td><td>21.20%</td><td>7.97</td><td>44.17</td><td>269</td></tr><tr><td>bL-ResNeXt-101 (32×4d) (α =2,β =4)</td><td>21.08%</td><td>4.08 (1.95×)</td><td>41.51</td><td>169 (1.59×)</td></tr><tr><td>bL-ResNeXt-101@256(α = 2, β=4)</td><td>20.48%</td><td>5.33 (1.50x)</td><td>41.51</td><td>203 (1.33×)</td></tr><tr><td>ResNeXt-101(64×4d)</td><td>20.73%</td><td>15.46</td><td>83.46</td><td>485</td></tr><tr><td>bL-ResNeXt-101 (64×4d) (α = 2,β =4)</td><td>20.48%</td><td>7.14 (2.17×)</td><td>77.36</td><td>263 (1.98×)</td></tr><tr><td>bL-ResNeXt-101@256 (64×4d) (α = 2, β = 4)</td><td>19.65%</td><td>9.32 (1.66x)</td><td>77.36</td><td>318 (1.53×)</td></tr><tr><td>SEResNeXt-50 (32×4d)</td><td>21.78%</td><td>4.23</td><td>27.56</td><td>216</td></tr><tr><td>bL-SEResNeXt-50 (32×4d) (α = 2,β=4) bL-SEResNeXt-50@256 (32×4d) (α= 2, β = 4)</td><td>21.44%</td><td>3.03 (1.40×)</td><td>28.77</td><td>163 (1.33×)</td></tr><tr><td></td><td>20.74%</td><td>3.95 (1.08×)</td><td>28.77</td><td>192 (1.03×)</td></tr><tr><td>SEResNeXt-101 (32×4d)</td><td>21.00%</td><td>7.97</td><td>48.96</td><td>376</td></tr><tr><td>bL-SEResNeXt-101 (32×4d) (α = 2, β = 4)</td><td>20.87%</td><td>4.08 (1.95×)</td><td>45.88</td><td>235 (1.60×)</td></tr><tr><td>bL-SEResNeXt-101@256 (32×4d) (α = 2,β= 4)</td><td>19.87%</td><td>5.33 (1.50×)</td><td>45.88</td><td>270 (1.39×)</td></tr></table>
+
+†: speed is benchmarked on NVIDIA Tesla K80 with batch size 16. Except for $@ 2 5 6$ , speed is evaluated under image size $2 2 4 \times 2 2 4$ . We trained all the ResNet, ResNeXt and SEResNeXt models by ourselves, so the accuracy is slightly different from the papers.
+
+ResNet as the backbone network. We experimented with different complexity control factors ( $\overset { \cdot } { \alpha }$ and $\beta$ ) to better understand their effects on performance. $\alpha$ and $\beta$ control both the structural and computational complexity of the Little-Branch, which determines the overall computational cost of bL-Net.
+
+As can be seen in Table 1, all the models based on ResNet-50 yield better performance over the baseline with less computation, clearly demonstrating the advantage of combining low- and highcomplexity networks to balance between speed and accuracy. In addition, the small performance gaps between these models suggest that a computationally light Little-Branch ( $\textless 1 5 \%$ of the entire network) can compensate well for the low resolution representation by providing finer image details. We consider $\alpha = 2$ and $\beta = 4$ as the default setting for the following experiments. Furthermore, there are more ablation studies on the design of bL-Net in the Appendix A.4.
+
+We further evaluated our approach on deeper models by using ResNet-101 and ResNet-152 as the backbone networks. We see from Table 2 that bL-ResNet-101 and bL-ResNet-152 behave similarly to bL-ResNet-50. As expected, both of them produce better results against the baseline models and achieving notable computational gains. Interestingly, our approach computationally favors deeper models, as evidenced by the fact that more speedups are observed on bL-ResNet-152 $( 2 . 3 \times )$ than on bL-ResNet-101 $( 2 . 0 \times )$ and bL-ResNet-50 $( 1 . 4 \times )$ . This is mainly because the Little Branch operating on low resolution spends less computation in a deeper model.
+
+ResNeXt and SEResNeXt as the backbone network. We extended bL-Net to ResNeXt and SEResNeXt, two of the more accurate yet compact network architectures. We also experimented with (SE)ResNeXt-50 and (SE)ResNeXt-101 using the $3 2 \times 4 \mathrm { d }$ and $6 4 \times 4 \mathrm { d }$ setting (Xie et al., 2017; Hu et al., 2018). In our case, the Big-Branch follows the same setting of (SE)ResNeXt; however, we changed the cardinality of the Little-Branch to align with the input channels of a group convolution in the Big-Branch. All the results are shown in Table 2.
+
+![](images/a6d6b75cec7913fe409b728001e78bc9467fe9e9c10ca41f278de9c663887d7f.jpg)  
+Figure 2: Comparison with the ResNet and ResNeXt related works.
+
+bL-ResNeXt-50 achieves a moderate speedup $( 1 . 4 0 \times )$ and provides an additional gain of $0 . 6 \%$ i n accuracy. However, bL-ResNeXt-101 $( 3 2 \times 4 \mathrm { d } )$ gains a much more substantial speedup of $2 \times$ and seeing $0 . 1 2 \%$ improvement in accuracy. The same trend can be seen on bL-ResNeXt-101 $( 6 4 \times 4 \mathrm { d } )$ . On the other hand, our bL-SEResNeXts also produce better performance while reducing more FLOPs than SEResNeXts. bL-SEResNeXt-101 achieves $2 \times$ speedups and improves accuracy by $0 . 1 3 \%$ .
+
+Since our bL-Net saves more budget in computation, we can evaluate a model at a larger image scale for better performance, e.g., $2 5 6 \times 2 5 6$ . As illustrated in Table 2, our models evaluated at $2 5 6 \times 2 5 6$ is consistently better than their corresponding baselines while still using fewer FLOPs. The advantage becomes more pronounced with deeper models. For instance, with $40 \%$ reduction on FLOPs, bL-ResNeXt-101 $\textcircled { \scriptsize { a } } 2 5 6$ $( 6 4 \times 4 \mathrm { d } )$ boosts the top-1 performance by $1 . 1 \%$ , which is quite impressive given that ResNeXt-101 $( 6 4 \times 4 \mathrm { d } )$ is a very competitive baseline. Table 2 also shows the running times of bL-Net on GPU, which indicate the practical speedups of these models are consistent with the theoretical FLOP reductions reported in Table 2.
+
+# 4.1.1 COMPARISON WITH RELATED WORK
+
+We first compared our method with the approaches that aim to accelerate ResNets or ResNeXts using techniques such as network pruning and adaptive computations. The results are shown in Figure 2.
+
+Our bL-Net significantly outperforms all related works regarding FLOPs reduction and accuracy. Our bL-ResNet-101 is $\sim 5 \%$ better than the network pruning approaches such as PFEC (Li et al., 2017) and LCCL (Dong et al., 2017), but still using less computation. When compared to SACT and ACT (Figurnov et al., 2017), our bL-ResNet-101 improves the accuracy by $5 \%$ while using the same number of FLOPs. On the other hand, our bL-ResNet-101 outperforms some of the most recent works including BlockDrop (Wu et al., 2018), SkipNet (Wang et al., 2018) and ConvNet-AIG (Veit & Belongie, 2018) by $3 . 7 \%$ , $2 . 2 \%$ , $1 . 2 \%$ top-1 accuracy at the same FLOPs, respectively. This clearly demonstrates the advantages of a simple fusion of two branches at different scales over the more sophisticated dynamic routing techniques developed in these approaches. In comparison to SPPoint (Kuen et al., 2018) under the same FLOPs, our bL-ResNet-101 surpasses it by $2 . 2 \%$ in accuracy.
+
+We also compared bL-Net with the variants of ResNets, like ResAttNe(X)t (Wang et al., 2017), SEResNeXt (Hu et al., 2018) and CBAM (Woo et al., 2018). These models introduces attention mechanisms to enhance feature representations in either the channel or the spatial domain. From Figure 2, it can be seen that our bL-Net outperforms all of them in both FLOPs and accuracy. Our bL-Net achieves better performance than ResAttNeXt while saving $1 . 5 \times \mathrm { F L O P s }$ . It also surpasses ${ \sim } 1 \%$ point with similar FLOPs or the similar accuracy with ${ \sim } 1 . 7 \times$ FLOPs reduction for both SEResNeXt and CBAM. It’s worth noting that bL-Net can be potentially integrated with these models to further improve their accuracy and efficiency.
+
+Finally, we ran a benchmark test on bL-ResNeXt $@ 2 5 6$ under the PyTorch framework with a batch size of 16 on a K80 GPU, and compared against various models that are publicly available. These models include Inception-V3 (Szegedy et al., 2016a), Inception-V4 (Szegedy et al., 2017), Inception-ResNetV2 (Szegedy et al., 2017), PolyNet (Zhang et al., 2017), NASNet (Zoph et al., 2018), PNASNet (Liu et al., 2018), DualPathNet (Chen et al., 2017b) and DenseNet (Huang et al., 2017b). Among them, NASNet currently achieves the best accuracy on ImageNet.
+
+![](images/d00d2c0fac8445379485651a3bc5072b0003cb35fce602124c3b6ac1929e4ff2.jpg)  
+Figure 3: Comparison performance among other types of networks. (a) FLOPs. (b) GPU Speed.
+
+From Fig. 3, we can see that overall, our bL-ResNeXt gains a better tradeoff between efficiency and accuracy. Compared to the Inception networks, bL-Net are better in both FLOPs and GPU running time. The bL-ResNeXt is also $2 \%$ point better than DenseNet at the same running speed, and $2 \times$ faster than DualPathNet at the same performance.
+
+NASNet achieves lower FLOPs and higher accuracy than bL-Net; however, the networks result in slow GPU running time since their operations are divided into small pieces (i.e. a fragmented structure), which are not friendly for parallel computations. On the other hand, bL-Net, although requiring more FLOPs, can still enjoy the computation optimization brought by modern GPU cards. Moreover, compared with the lowest FLOP configuration of NASNet, our bL-ResNeXt is $0 . 5 \%$ point better while running $1 . 5 \times$ faster.
+
+To further validate the generality of bL-Net, we also integrated the bL-Net with the highly efficient network, ShuffleNetV2 (Ma et al., 2018), and the results can be found in Appendix A.5. We also demonstrate the adaptability of $b L$ -Net on the object detection task (see Appendix A.6).
+
+# 4.2 SPEECH RECOGNITION
+
+We train ResNet style acoustic models in the hybrid framework on Switchboard $+$ Fisher (2000h) and provide results on Hub5 (Switchboard and Call Home portions). Switchboard is a large dataset with 2000 hours of transcribed speech from 28, 000 speakers, which is actively used as benchmark (Xiong et al., 2016; Saon et al., 2017) akin to ImageNet in the computer vision community. Our ResNet acoustic models are similar to the state of the art models described in (Saon et al., 2017), though slightly simplified (less fully connected layers) and trained with a simpler procedure (no class balancing). We provide results only after Cross-Entropy training and after decoding with a small language model (4M n-grams). Gains from this setting are typically maintained in the standard further pipelines like fine-tuning with sequence training, using more complex language models.
+
+Appendix B gives a thorough overview of the architecture of the speech acoustic models. The main difference speech acoustic models have compared to image classification networks, is that striding or pooling only happens along the frequency axis, while along in the time direction we need to output dense predictions per frame (Sercu & Goel, 2016). This means that the branches at different resolutions have a fundamentally different view of the signal as it is propagating through the network; the ratio of resolution in frequency (downsampled in the Big-Branch) vs resolution in time (same between branches) is different. We can think about this as the convolutional kernels having different “aspect ratios” between branches. Therefore we not only expect FLOP reductions in bL-Net, but expect to have increased representational power. In addition, similar to the case in object recognition (Table 2), we could process the speech signal at higher frequency resolution than what is computationally feasible for the baseline ResNets.
+
+Table 3 shows the results for the different architectures described in Appendix B. Most results are in line with the observations in the object recognition bL-Net. When comparing the baseline ResNet-22 (line 1) to the best bL-ResNet-22 (line 5), we see not only a reduction in FLOPs, but also a modest gain in Word Error Rate (WER). Comparing lines 2-4, we see that increasing $\beta$ (i.e. shorter little branches at full resolution) causes no WER degradation, while reducing the number of FLOPs. From line 5 we see that, similar to the object recognition ResNet results, decreasing $\alpha$ from 4 to 2 (i.e. keeping more feature maps in the full-resolution little branches) is important for performance, even though this increases the FLOPs again. We can summarize the best setting of $\alpha = 2$ and $\beta = 3$ for the little branches at full resolution: make them shorter but with more feature maps. This is consistent with the image classification results. From line 2 vs. line 6, the concatenation merge mode performs similar to the default additive merging, while increasing the number of FLOPs. Line 7 (compare to line 2) shows an experiment with additional branches on the lower layers (See Appendix B). Although there is some gain in WER, the added parameters and compute on the lower layers may not make this a worthwhile trade-off.
+
+Table 3: Speech recognition results. We present results on Hub5 and the CallHome portion of Hub5, while the RT-02 Switchboard set was used for selecting decode epoch and HMM prior settings.   
+
+<table><tr><td>Model</td><td></td><td>FLOPs (109)</td><td>Params (106)</td><td>WER Avg</td><td>Hub5</td><td>Hub5 CH</td></tr><tr><td>1</td><td>Baseline: ResNet-22</td><td>1.11</td><td>3.02</td><td>14.67%</td><td>11.15%</td><td>18.17%</td></tr><tr><td>2</td><td>bL-ResNet-22 (α = 4,β = 1)</td><td>0.68 (1.63×)</td><td>3.15</td><td>14.72%</td><td>11.24%</td><td>18.18%</td></tr><tr><td>3</td><td>bL-ResNet-22 (α = 4,β = 2)</td><td>0.66 (1.68×)</td><td>3.11</td><td>14.47%</td><td>10.95%</td><td>17.95%</td></tr><tr><td>4</td><td>bL-ResNet-22(α = 4,β= 3)</td><td>0.65 (1.70×)</td><td>3.10</td><td>14.66%</td><td>11.25%</td><td>18.05%</td></tr><tr><td>5</td><td>bL-ResNet-22 (α = 2,β = 3)</td><td>0.77 (1.43×)</td><td>3.07</td><td>14.46%</td><td>11.10%</td><td>17.80%</td></tr><tr><td>6</td><td>bL-ResNet-22 (α = 4,β=1) cat</td><td>0.70 (1.58×)</td><td>3.18</td><td>14.67%</td><td>11.31%</td><td>18.00%</td></tr><tr><td>7</td><td>bL-PYR-ResNet-22 (α = 4, β =1)</td><td>0.98 (1.13×)</td><td>3.32</td><td>14.50%</td><td>11.05%</td><td>17.92%</td></tr></table>
+
+# 4.3 DISCUSSION ON bL-Net
+
+From the results of both tasks, we observe the following common insights, which enable us to design an efficient multi-scale network with competitive performance: (I) The Little-Branch can be very light-weight, (II) bL-Net performs better when the Little-Branch is wide and shallow (smaller $\alpha$ and larger $\beta$ ), (III) merging is effective when the feature dimension has changed, and (IV) branch merging by addition is more effective than concatenation. (I) is because the Big-Branch can extract essential information, a light Little-Branch is good enough to provide sufficient information the Big-Branch lacks. Regarding (II), wider networks have been shown to perform better than deep networks while using a similar number of parameters. (III) is well-discussed in Appendix A.4. Finally (IV), merging through addition provides better regularization for both branches to learn complementary features to form strong features.
+
+# 5 CONCLUSION
+
+We proposed an efficient multi-scale feature representation based on integrating multiple networks for object and speech recognition. The Big-Branches gain significant computational reduction by working at low-resolution input but still extract meaningful features while the Little-Branch enriches the features from high-resolution input but with light computation. On object recognition task, we demonstrated that our approach provides approximately $2 \times$ speedup over baselines while improving accuracy, and the result significantly outperforms the state-of-the-art networks by a large margin in terms of accuracy and FLOPs reduction. Furthermore, when using the proposed method on speech recognition task, we gained $0 . 2 \%$ WER and saved $30 \%$ FLOPs at the same time. In pratice, the proposed bL-Net shows that the reduced FLOPs can consistently speed up the running time on GPU. That evidence showed that the proposed bL-Net is an efficient multi-scale feature representation structure for competitive performance with less computation. In this paper, we chose ResNet, ResNeXt and SEResNeXt as our backbone networks but $b L$ -Net can be integrated with other advanced network structures, like DenseNet (Huang et al., 2017b), DualPathNet (Chen et al., 2017b) and NASNet (Zoph et al., 2018) to achieve competitive performance while saving computations. Furthermore, bL-Net can be integrated with those CNN acceleration approaches to make models more compact and efficient.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank Dr. Paul Crumley and Dr. I-Hsin Chung for their help with the hardware infrastructure setup.
+
+Quanfu Fan and Rogerio Feris are supported by IARPA via DOI/IBC contract number D17PC00341. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DOI/IBC, or the U.S. Government.
+
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+# Appendix
+
+The appendix illustrates the details of our experiments on object recognition and speech recognition and more ablation study.
+
+# A bL-Net FOR OBJECT RECOGNITION
+
+# A.1 EXPERIMENTAL SETUP
+
+We used the ImageNet dataset for all experiments. We trained all the models by Tensorpack (Wu, 2017), a higher-level wrapper for Tensorflow (Abadi et al., 2015). All the models were trained with 110 epochs, batch size 256, weight decay 0.0001, momentum 0.9 and Nesterov momentum optimizer. Furthermore, we used cosine learning-rate schedule as (Huang et al., $2 0 1 7 \mathrm { a }$ ; Loshchilov & Sgdr, 2017). We deployed the popular augmentation technique in (Szegedy et al., 2016b; Gross & Wilber, 2016) to increase the variety of training data, and randomly crop a $2 2 4 \times 2 2 4$ patch as training image. The validation error is evaluated by resizing the shorter side of an image to 256 and then crop a $2 2 4 \times 2 2 4$ from the center. Note that the results reported here for the vanilla ResNet, ResNeXt and SEResNeXt models are difference from those reported in the original paper (He et al., 2016; Gross & Wilber, 2016; Hu et al., 2018). Our vanilla ResNet is better than the original paper while vanilla ResNeXt and SEResNeXt is slightly worse than the original paper.
+
+Table 4: Network configurations of bL-ResNet-50. Output size is illustrated in the parenthesis.   
+
+<table><tr><td rowspan=1 colspan=1>Layers</td><td rowspan=1 colspan=1>bL-ResNet-50</td><td rowspan=1 colspan=1>ResNet-50</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=2>7 × 7,64, s2 (112 × 112)</td></tr><tr><td rowspan=1 colspan=1>bL-module</td><td rowspan=1 colspan=1>3×3,323 × 3,64,s2         3×3,32,s2(56 × 56)(1×1,64</td><td rowspan=1 colspan=1>MaxPooling (56 × 56)</td></tr><tr><td rowspan=1 colspan=1>bL-module</td><td rowspan=1 colspan=1>ResBlockg,256 ×2      ResBlockL,128 ×1ResBlock,256,s2 (28 × 28)</td><td rowspan=1 colspan=1>ResBlock,256 ×3, s1(56 × 56)</td></tr><tr><td rowspan=1 colspan=1>bL-module</td><td rowspan=1 colspan=1>ResBlockp,512 ×3      ResBlockL,256 ×1ResBlock, 512, s2 (14 × 14)</td><td rowspan=1 colspan=1>ResBlock,512 ×4, s2(28× 28)</td></tr><tr><td rowspan=1 colspan=1>bL-module</td><td rowspan=1 colspan=1>ResBlockb,1024 ×5     ResBlockL,512 ×1ResBlock,1024 (14 × 14)</td><td rowspan=1 colspan=1>ResBlock,1024 ×6, s2(14 × 14)</td></tr><tr><td rowspan=1 colspan=1>ResBlock</td><td rowspan=1 colspan=2>ResBlock,2048 ×3, s2 (7 × 7)</td></tr><tr><td rowspan=1 colspan=1>Average pool</td><td rowspan=1 colspan=2>7 ×7 average pooling</td></tr><tr><td rowspan=1 colspan=1>FC, softmax</td><td rowspan=1 colspan=2>1000</td></tr></table>
+
+ResBlockB: the first $3 \times 3$ convolution is with stride 2, and a bi-linear upsampling is applied at the end. $\mathbf { R e s B l o c k } _ { L }$ : a $1 \times 1$ convolution is applied at the end to align the channel size. $\mathbf { s } 2$ : the stride is set to 2 for the $3 \times 3$ convolution in the ResBlock.
+
+# A.2 NETWORK STRUCTURE
+
+This section shows the details of network structures of our bL-ResNet, and the setting of $\alpha$ and $\beta$ is 2 and 4, respectively. To understand how do we design bL-ResNet-50 based on ResNet-50, Table 4 shows the details of network structure. We used a bottleneck residual block as a ResBlock, and a ResBlock, $C$ denotes a block composed of $1 \times 1 , 3 \times 3$ , and $1 \times 1$ convolutions, where the first $1 \times 1$ and the $3 \times 3$ have $C / 4$ kernels and the last $1 \times 1$ has $C$ kernels.
+
+First, the Big-Branch and the Little-Branch shares a residual block at the transition layer, so the number of residual blocks in each branch will be subtracted by 1. The number of residual blocks in the Little-Branch is defined as $\begin{array} { r } { \left\lceil { \frac { L } { \beta } } \right\rceil - 1 } \end{array}$ and at least one, where $L$ is the number of residual blocks in the Big-Branch, and the number of kernels in a convolutional layer would be $\frac { C } { \alpha }$ , where $C$ is the number of kernels in the Big-Branch. Thus, for all stages, the number of blocks in the Little-Branch is only one, and the number of blocks in the Big-Branch would be the number of blocks in ResNet-50 miuns one.
+
+For bL-ResNet-101 and bL-ResNet-152, we redistributed the residual blocks to different stages to balance the residual blocks at each stage. A ResNet model has 5 stages, and each stage has the same spatial size. These two models accumulate most of the convolutions (or computations) on the $4 ^ { t h }$ stage, where the size of feature maps is $1 4 \times 1 4$ when input size is $2 2 4 \times 2 2 4$ . While such a design may be suitable for a very deep model, it likely limits the ability of Big-Branch to learn information at large scales, which mostly resides at earlier stages. Thus, we move some blocks in the $4 ^ { t h }$ stage of these two models to the ${ \dot { \mathbf { \zeta } } } _ { 2 ^ { n d } }$ and $3 ^ { r d }$ stages. Table 5 shows the details of bL-ResNet-101 and bLResNet-152.
+
+Table 5: Network configurations of bL-ResNets, and $\alpha = 2$ and $\beta = 4$   
+
+<table><tr><td>Layers</td><td>Output Size</td><td colspan="6">bL-ResNet-101 bL-ResNet-152</td></tr><tr><td>Convolution</td><td>112 × 112</td><td colspan="6">7 × 7,64,s2</td></tr><tr><td>bL-module</td><td>56× 56</td><td colspan="6">（3×3,32 3 ×3,64,s2 3×3,32,s2 （1×1,64</td></tr><tr><td>bL-module</td><td>56×56</td><td>/1×1,64 3×3,64 ×3(2) (1×1,256) B</td><td>/1×1,32 3×3,32 （1×1,128） L</td><td>×1</td><td>/1×1,64 3×3,64 × 4(2) （1×1,256) B</td><td>/1×1,32 3×3,32 （1×1,128)</td><td>×1 L</td></tr><tr><td>transition layer</td><td>28×28</td><td colspan="6">/1×1,64 3×3,64,s2 ×1 （1×1,256</td></tr><tr><td>bL-module</td><td>28×28</td><td>(1×1,128) /1×1,64 3×3,128 ×7(3) 3×3,64 (1×1,512) B</td><td>×1 （1×1,256） L</td><td colspan="2">(1×1,128) 3×3,128 (1×1,512)</td><td>× 11(7)</td><td>/1×1,64 3×3,64 ×2 (1×1,256) L</td></tr><tr><td>transition layer</td><td>14 × 14</td><td colspan="2"></td><td colspan="2">/1×1,128 3×3,128,s2 ×1 (1×1,512</td><td colspan="3">B</td></tr><tr><td>bL-module</td><td>14 × 14</td><td>/1×1,256 3×3,256 �� 17(22) （1×1,1024）B</td><td>(1×1,128) 3×3,128 （1×1,512 L</td><td>×3</td><td>/1×1,256 3×3,256 (1×1,1024) B</td><td>× 29(35)</td><td>(1×1,128) 3×3,128 （1×1,512）</td><td>×6 L</td></tr><tr><td>transition layer</td><td>14 × 14</td><td colspan="6">/1×1,256 3×3,256 ×1 （1×1,1024）</td></tr><tr><td>ResBlock</td><td>7×7</td><td colspan="6">/1×1,512 3 ×3,512,s2 ×3</td></tr><tr><td>Average pool</td><td>1×1</td><td colspan="6">（1×1,2048 7×7 average pooling</td></tr><tr><td>FC, softmax</td><td colspan="7">1000</td></tr></table>
+
+For each $\overline { B }$ block, the first $3 \times 3$ convolution is with stride 2, and a bi-linear upsampling is applied at the end. For each $L$ block, a $1 \times 1$ convolution is applied at the end. $^ { s 2 }$ : the stride is set to 2 for the convolutional layer. The number in the parethesis denotes the original number of blocks in ResNet.
+
+# A.3 PERFORMANCE ON LOW RESOLUTION INPUT
+
+We analyzed what advantages bL-Net could provide as compared to the network which works on low resolution input directly (ResNet-50-lowres). As shown in Table 6, ResNet-50-lowres reduces lots of computations but its accuracy is not acceptable; however, bL-ResNet-50 $\alpha = 2$ and $\beta = 4$ ) achieves a better balance between accuracy and performance. A similar trend is also observed on a deeper model ResNet-101-lowres. While such performance is unsatisfying compared to the state of the art, it is quite reasonable and expected given that almost $3 \sim 4 \times$ reduction of computation are achieved in such a case.
+
+Figure 4 shows the prediction results from bL-ResNet-50 and ResNet-50-lowres. When both models predict correctly (4 (a) and (b)), the bL-ResNet-50 provides better confidence for the prediction. Because the object only occupies a small portion of an image, the Little-Branch can still capture the object clearly. On the other hand, when the key features of an object is small, like the shape of beak of a bird (c) and the spots of a ladybug (d), bL-ResNet-50 can easily retain that key feature to predict correctly while ResNet-50-lowres provides wrong predicted label.
+
+![](images/bf948a2c1696a33afbd37fc5e16cb47be047e9384139f171958c590313dc30a9.jpg)  
+Figure 4: Prediction results for bL-ResNet-50 and ResNet-50-lowres. True labels, predicted labels and their probability are listed in the table. When both models predicts correctly ((a) and (b)), bL-ResNet-50 achieves much higher probability; on the other hand, bL-ResNet-50 captures the details on the object and then predicts correctly ((c) and (d)).
+
+Table 6: Performance of ResNets at different input resolutions.   
+
+<table><tr><td>Network</td><td>Top-1 Error</td><td>FLOPs (109)</td><td>Params (106)</td></tr><tr><td>ResNet-50</td><td>23.66%</td><td>4.09</td><td>25.55</td></tr><tr><td>ResNet-50-lowres</td><td>26.10%</td><td>1.29 (3.17×)</td><td>25.60</td></tr><tr><td>ResNet-101</td><td>21.95%</td><td>7.80</td><td>44.54</td></tr><tr><td>ResNet-101-lowres</td><td>24.80%</td><td>2.22 (3.51x)</td><td>44.57</td></tr></table>
+
+# A.4 ABLATION STUDY ON NETWORK MERGING AND MULTI-BRANCH
+
+Is linear combination better than concatenation? We adopt the simpler addition in bL-Net. Nonetheless, if we design the Big-Branch in a way that the output channels is identical to the backbone networks, then the number of kernels in the Big-Branch would be only $1 - \alpha$ with respect to the total number of kernels of the backbone network; thus, in this case, the overall $b L$ -Net can be more efficient while the performance degradation could be compromised. We compared the performance of these two different merging schemes in Table 7. Although concatenation approach is more efficient, it performs much worse than addition with a gap of almost $1 . 5 \%$ . This leaves addition as a better choice for bL-Net in both visual and speech tasks.
+
+More-branch in bL-Net As mentioned in Section 3.1, our approach can be extended to a scenario with multiple image scales. We experimented with three scales [1/4, 1/2, 1] on bL-ResNet-50 $( K = 3$ ) where ResNet-50 is served as the Big-Branch at the scale of 1/4 of the original input, i.e. $5 6 \times 5 6$ As indicated in Table 7, a 3-scale bL-Net requires more FLOPs and parameters due to the fact that the overhead in merging more branches is significant for ResNet-50, but even though, it still cannot provide superior performance of a 2-scale bL-Net. This is because the Big-Branch in the 3-scale bL-Net is downsampled aggressively by 4 times, thus substantially degrade the capability of feature representation in the Big-Branch.
+
+Table 7: Different scales and merging schemes on bL-ResNet.   
+
+<table><tr><td>Network</td><td>Top-1 Error</td><td>FLOPs (109)</td><td>Params (106)</td></tr><tr><td>ResNet-50</td><td>23.66%</td><td>4.09</td><td>25.55</td></tr><tr><td>bL-ResNet-50 (addition,K = 2)</td><td>22.69 %</td><td>2.85 (1.43×)</td><td>26.69</td></tr><tr><td>bL-ResNet-50 (concatenation,K = 2)</td><td>24.04%</td><td>2.01 ( (2.03x)</td><td>20.57</td></tr><tr><td>bL-ResNet-50 (addition,K = 3)</td><td>24.12%</td><td>3.91 (1.04×)</td><td>27.23</td></tr></table>
+
+Table 8: Different number of merges in bL-ResNet. $m$ : number of merges. $( \alpha = 2 , \beta = 4 )$   
+
+<table><tr><td>Model</td><td>Top-1 Error</td><td>FLOPs (109)</td><td>Params (106)</td></tr><tr><td>bL-ResNet-50 (m = 4) (baseline)</td><td>22.69%</td><td>2.85</td><td>26.69</td></tr><tr><td>bL-ResNet-50 (m = 2)</td><td>23.48%</td><td>2.74</td><td>26.66</td></tr><tr><td>bL-ResNet-50 (m = 1)</td><td>24.57%</td><td>2.64</td><td>26.64</td></tr><tr><td>bL-ResNet-101 (m= 4) (baseline)</td><td>21.80%</td><td>3.89</td><td>41.85</td></tr><tr><td>bL-ResNet-101 (m = 7)</td><td>21.85%</td><td>5.21</td><td>44.44</td></tr></table>
+
+# Number of Merges in bL-Net
+
+We also analyzed the number of merges we needed in the bL-Net. One big difference between our approach and others is that $b L$ -Net merges multiple times as opposed to only once in most of the other approaches. Below we provide an explanation of why more information exchange is encouraged in our approach and when is the best moment for merging operation.
+
+In the above bL-Net, we merged branches before the feature dimension changes, except for the first stride convolution; thus, we used 4 merges $m = 4 ,$ ). We experimented with a different number of merges for bL-ResNet-50 and bL-ResNet-101, and the results are shown in Table 8. Since there are fewer layers in bL-ResNet-50, we reduce the number of merges to show their importance; on the other hand, there are more layers in bL-ResNet-101, so we add more merges to show that those additional merges would similarly not improve the performance anymore.
+
+The accuracy of the bL-ResNet-50 models with less number of merges ( $m = 1$ and $m = 2$ ) is significantly worse than with more $\textlangle m = 4$ ) and they do not save many FLOPs and parameters at all. This justifies frequent information exchange improves the performance. On the other hand, bL-ResNet-101 $m = 7 ,$ ) uses more merges; however, it also does not improve the performance and requires more FLOPs, which comes from more merges. This is because the original setting for the amount of merging happened when either the channel number or feature map size is changed, so extra merges happened at the feature dimension. Thus, those extra merges could be redundant since merging at identical dimension could be reduced to one merging. Hence, it empirically proves that merging before dimension is changed is the most effective.
+
+# A.5 bL-Net FOR HIGH-EFFICIENCY NETWORK
+
+To demenstrate bL-Net can be applied on different types of network, we deploy bL-Net on the high-efficiency network, ShuffleNetV2 (Ma et al., 2018), and results are shown in Table 9. Our bL-ShuffleNetV2@256 outperforms ShuffleNetV2 by up to $0 . 5 \%$ point under similar FLOPs, which suggests that our approach can also improve the high-efficiency networks.
+
+Table 9: Comparison with ShuffleNetV2 (Ma et al., 2018) $( \alpha = 2 , \beta = 2 ) .$   
+
+<table><tr><td>Model</td><td>|Model width</td><td>Top-1 Error</td><td>FLOPs (106)</td><td>Params (106)</td></tr><tr><td rowspan="3">ShuffleNetV2</td><td>1×</td><td>30.60%</td><td>146</td><td>2.30</td></tr><tr><td>1.5×</td><td>28.03%</td><td>299</td><td>3.51</td></tr><tr><td>2×</td><td>26.85%</td><td>588</td><td>7.40</td></tr><tr><td rowspan="3">bL-ShuffleNetV2 @ 256</td><td>1×</td><td>30.84%</td><td>150</td><td>2.33</td></tr><tr><td>1.5×</td><td>27.83%</td><td>298</td><td>4.80</td></tr><tr><td>2×</td><td>26.38%</td><td>590</td><td>7.60</td></tr></table>
+
+All models are trained by ourselves.
+
+# A.6 bL-Net FOR OBJECT DETECTION
+
+We demonstrate the effectiveness of bL-Net on object detection. We use bL-Net as a backbone network for FasterRCNN $+$ FPN (Lin et al., 2017) on the PASCAL VOC (Everingham et al., 2010) and MS COCO datasets (Lin et al., 2014). Table 10 shows the comparison with the detector with ResNet-101
+
+as the backbone network, and the results show that bL-Net achieves competitive performance while saving about $1 . 5 \times \mathrm { F L O P s } ^ { 1 }$ , suggesting that bL-Net is transferable to other vision tasks.
+
+Table 10: Objection detection results. Detection performance on the PASCAL VOC 2007test dataset.   
+
+<table><tr><td>Network</td><td>mAP@[IoU=0.5] (bbox)</td><td>FLOPst (109)</td></tr><tr><td>ResNet-101</td><td>81.5</td><td>137.70</td></tr><tr><td>bL-ResNet-101 (α=2, β=4)</td><td>81.4</td><td>89.21 (1.54×)</td></tr><tr><td>bL-ResNet-101 (α=2, β=2)</td><td>81.5</td><td>93.95 (1.47×)</td></tr></table>
+
+Detection performance on the MS COCO val2017 dataset.   
+
+<table><tr><td>Network</td><td>mAP@[IoU=0.50:0.95](bbox)|FLOPs‡ (109)</td><td></td></tr><tr><td>ResNet-101</td><td>39.2</td><td>234.85</td></tr><tr><td>bL-ResNet-101 (α=2, β=4)</td><td>39.5</td><td>151.11 (1.55×)</td></tr><tr><td>bL-ResNet-101 (α=2, β=2)</td><td>40.4</td><td>160.21 (1.47×)</td></tr></table>
+
+†: FLOPs is calculated when the size of input image is $6 0 0 \times 1 0 2 4$ with 300 proposals. ‡: FLOPs is calculated when the size of input image is $8 0 0 \times 1 3 4 4$ with 300 proposals.
+
+# A.7 COMPARISON OF MEMORY REQUIREMENT
+
+We benchmarked the GPU memory consumption in runtime at both the training and test phases for all the models evaluated in Fig. 3. The results are shown in Fig. 5. The batch size was set to 8, which is the largest number allowed for NASNet on a P100 GPU card. The image size for any model in this benchmark experiment is the same as that used in the experiment reported in Fig. 3. For bL-Net, the input image size is $2 2 4 \times 2 2 4$ in training and $2 5 6 \times 2 5 6$ in test.
+
+From Fig. 5, we can see that bL-Net is the most memory-efficient for training among all the approaches. In test, bL-ResNeXt consumes more memory than inception-resnet-v2 and inception-v4 at the same accuracy, but bL-SEResNeXt outperforms all the approaches. Note that NASNet and PNASNet are not memory friendly. This is largely because they are trained on a larger image size $( 3 3 1 \times 3 3 1 )$ ) and these models are composed of many layers.
+
+![](images/da4e5aec7cfa4b70d0b83baf77d263cacc8fcabedbed9c1b250d87d538f80742.jpg)  
+Figure 5: Comparison memory requirement at the training and test phases among other types of networks. (a) Training. (b) Test. (The batch size is 8.)
+
+Table 11: Complexity study of the Little-Branch $\scriptstyle { \alpha }$ and $\beta$ ) for bL-ResNet-50 and bL-ResNet-101.   
+
+<table><tr><td>Model</td><td>Top-1 Error</td><td>FLOPs (109)</td><td>Params (106)</td></tr><tr><td>ResNet-50</td><td>23.66%</td><td>4.09</td><td>25.55</td></tr><tr><td>bL-ResNet-50 (α =1, β=1)</td><td>21.75%</td><td>5.65</td><td>34.12</td></tr><tr><td>bL-ResNet-50 (α =1, β=2）</td><td>22.11%</td><td>4.34</td><td>30.14</td></tr><tr><td>bL-ResNet-50(α= 2,) β=2)</td><td>22.72%</td><td>2.91</td><td>26.97</td></tr><tr><td>bL-ResNet-50(α =2,) β=4)</td><td>22.69%</td><td>2.85</td><td>26.69</td></tr><tr><td>bL-ResNet-50(α = 4, β = 2)</td><td>23.20%</td><td>2.49</td><td>26.31</td></tr><tr><td>bL-ResNet-50 (α =4, β= 4)</td><td>23.15%</td><td>2.48</td><td>26.24</td></tr><tr><td>ResNet-101</td><td>21.95%</td><td>7.80</td><td>44.54</td></tr><tr><td>bL-ResNet-101 (α=1,β=1)</td><td>20.31%</td><td>10.29</td><td>63.32</td></tr><tr><td>bL-ResNet-101 (α = 2, β= 2)</td><td>21.40%</td><td>4.27</td><td>43.39</td></tr><tr><td>bL-ResNet-101 (α = 2,β=4)</td><td>21.80%</td><td>3.89</td><td>41.85</td></tr></table>
+
+# B bL-Net FOR SPEECH RECOGNITION
+
+# B.1 EXPERIMENTAL SETUP
+
+In our experiments, we start with an input size of $6 4 \times 4 9$ , where 64 is the number of logmel filterbanks, calculated for each utterance on-the-fly. We also stack their first and second derivatives to get 3 input channels, resulting in our final input of dimensionality batch_si $\mathsf { \Omega } : \in \times 3 \times 6 4 \times 4 9$ . Our output is of size batch_si $\mathsf { z e } \times 5 1 2 \times 4 \times 1$ , which is then projected to batch_si $\mathsf { z e } \times 5 1 2 \times 1 \times 1$ and finally to batch_ $\mathrm { ~ s ~ i ~ z ~ e ~ } \times \mathrm { 3 2 k } \times 1 \times 1$ for classification. We then perform softmax cross-entropy over this output space of 32k tied CD states from forced alignment, doing phone prediction on the central frame of the input utterance. We report results after Cross-Entropy training, on ${ \mathrm { H u b } } 5 ^ { \prime } { 0 0 }$ (SWB and CH part) after decoding using the standard small 4M n-gram language model with a $3 0 . 5 \mathrm { k }$ word vocabulary.
+
+All models were trained in PyTorch (Paszke et al., 2017) over 16 epochs on 2 GPUs with per-GPU batch size 256 (total batch size of 512), gradient clipping 10.0, weight decay √ $1 \times 1 0 ^ { - 6 }$ , nesterov accelerated momentum 0.9, and learning rate 0.03 (annealed by $\sqrt { 0 . 5 }$ per epoch $\geq 1 0 $ ).
+
+# B.2 NETWORK STRUCTURES
+
+ResNet-22 Our models follow a ResNet architecture without padding in time (Saon et al., 2017), which accounts for the fact that padding in time adds undesirable artifacts when processing a longer utterance (Sercu & Goel, 2016). Under this constraint, each convolution operation reduces our input sequence in time by $k - 1$ , where $k$ is the kernel width used. This effect can be seen in Table 12, in which the time variable, $T$ , is reduced in accordance with the number of convolutions. For similar reasons, when we stride we only do so in frequency, and not in time. For the rest of this section, when we refer to striding we are referring only to striding in frequency. We define our residual blocks as a series of $3 \times 3$ convolutions. When we transition from one stage to the next, we stride by 2 on the first convolution of the following block. All of our models start with a $5 \times 5$ convolution with stride 2, which downsamples the input from 64 melbins to 32.
+
+Our baseline model, ResNet-22, consists of four stages of two-convolution residual blocks: $( 3 \times$ $3 , 6 4 ) \times 3$ $\times 3 ; ( 3 \times 3 , 1 2 8 , s 2 ) \times 3 ; ( 3 \times 3 , 2 5 6 , s 2 ) \times 3 ; ( 3 \times 3 , 5 1 2 , s 2 ) \times 2$ . The output then goes through a bottleneck projection layer to the 32k-dimensional output CD state: $( 4 \times 1 , 5 1 2 )$ and $( 1 \times 1 , 3 2 \mathbf { k } )$ .
+
+bL-ResNet-22 $( \alpha = 4 , \beta = 1 ,$ ) Table 12 displays two bL-Net architectures that we experimented with based on the ResNet-22 baseline. The bL-Net baseline, bL-ResNet-22 $( \alpha = 4 , \beta = 1 )$ , consists of two branches in each Big-Little Module and is well-defined through the parameters $\alpha$ and $\beta$ . In between each Big-Little Module we downsample our input using a transition layer consisting of a residual block with a single $3 \times 3$ convolution with stride 2. Therefore, we shift one convolution operation out of the last residual block in each stage that precedes a transition layer.
+
+Whenever downsampling is performed in the Big-Branch, it is only in the frequency dimension and not in time. Similarly, bilinear upsampling only occurs in the frequency dimension. All $b L$ -Net variants end with the same projection and output layer as ResNet-22. All merges, unless otherwise specified, are through linear combination with unit weights per branch. For comparison, we experimented with a version of bL-ResNet-22 $\mathbf { \Phi } ( \alpha = 4 , \beta = 1 $ ) using concatenation to merge branches, the results of which are presented in Table 3. Using concatenation instead of linear combination in this model results in each stage having more channels after concatenation than the current stage of the network calls for (i.e. in stage 1 we end up with $6 4 + 1 6 = 8 0$ channels at the end of the relevant Big-Little Module, whereas we only want 64 channels to be outputted). To resolve this, we apply a $1 \times 1$ convolution to reduce the number of channels accordingly and fuse the two separate feature maps.
+
+Table 12: Network configurations of bL-ResNets applied to speech for acoustic modeling.   
+
+<table><tr><td>Layers</td><td>Output Size 一</td><td colspan="3">bL-ResNet-22 (α=4,β=1)</td><td colspan="3">bL-ResNet-22 (α= 2,β=3)</td></tr><tr><td>Convolution</td><td>32×T T=49→45</td><td colspan="6">5×5,64,s2</td></tr><tr><td>bL-module</td><td>32×T T=45→35</td><td>（3×3,64) 3×3,64 ×2 B (3×3,64)g ×1</td><td>(3×3,16) （3×3,16）L (3 ×3,16), × 1</td><td>×2</td><td>（3×3,64) ×2 3×3,64) B (3 ×3,64)B ×1</td><td>（3×3,32） （3×3,32)</td><td>×1</td></tr><tr><td>transition layer</td><td>16×T T=35→33</td><td colspan="6">(3×3,64,s2)×1</td></tr><tr><td>bL-module</td><td>16×T T= 33→23</td><td>（3×3,128) ×2 （3×3,128） B</td><td>（3×3,32) （3×3,32）</td><td>×2 L</td><td>（3×3,128) ×2 3×3,128 B</td><td>（3×3,64) 3×3,64）</td><td>×1 L</td></tr><tr><td>transition layer</td><td>8×T T=23→21</td><td colspan="6">（3×3,128)B ×1 (3×3,32) ×1 （3×3,128)g ×1 （3×3,128,s2)×1</td></tr><tr><td>bL-module</td><td>8×T T=21→11</td><td>（3×3,256) ×2 3×3,256 B</td><td>(3×3,64) （3×3,64） L</td><td>×2</td><td>（3×3,256) ×2 3×3,256 B</td><td>（3×3,128) 3×3,128）</td><td>×1 L</td></tr><tr><td>transition layer</td><td>4×T</td><td colspan="6">(3×3,256)B ×1 (3×3,64) ×1 (3×3,256)g ×1 (3×3,256,s2) ×1</td></tr><tr><td>bL-module</td><td>T=11→9 4×T</td><td colspan="6">（3×3,512) （3×3,128) （3×3,512) ×2 ×2 ×2</td></tr><tr><td>Convolution</td><td>T=9→1 1×1</td><td colspan="6">（3×3,512） （3×3,128） （3×3,512) B L</td></tr><tr><td>Convolution</td><td>1×1</td><td colspan="6">4×1,512 1 × 1,32k</td></tr></table>
+
+For each $B$ block, the first $3 \times 3$ convolution is with stride 2 (in frequency), and a bilinear upsampling is applied at the end. For each $L$ block, a $1 \times 1$ convolution is applied at the end to match feature maps. $s 2$ : the stride is set to 2 in the frequency dimension (not in time) for the convolutional layer. $T = T _ { 0 }  T _ { 1 }$ indicates that $T _ { 0 }$ is the size of the time dimension at the start of the given layer, which is reduced to $T _ { 1 }$ by the end of the layer.
+
+bL-ResNet-22 $( \alpha = 4 , \beta = 2 , 3 )$ We explored two more models where we fix $\alpha = 4$ , one in which we take $\beta = 2$ and another where $\beta = 3$ . All Big-Branchs in these models are the same as bL-ResNet-22 $( \alpha = 4 , \beta = 1 )$ ). The difference in each Little-Branch is based on the setting of $\beta$ . Since the number of convolutions we use in the bL-Net baseline is uneven in the first three stages, we take $\lceil L / \beta \rceil$ to be the depth of the Little-Branch, where $L$ is the depth of the Big-Branch. For $\beta = 2$ , the first three stages of the network have a Little-Branch consisting of one residual block with two $3 \times 3$ convolutions and one residual block with one $3 \times 3$ convolution. This results in a reduction of the number of convolutions from 5 in the Big-Branch to 3 in the Little-Branch. For $\beta = 3$ , the first three stages of the network have a Little-Branch consisting of one residual block with two $3 \times 3$ convolutions, resulting in a reduction in the number of convolutions from 5 in the Big-Branch to 2 in the Little-Branch. For both models, the Little-Branch in the final stage consists of a single residual block with two $3 \times 3$ convolutions, since there are only four convolutions in the Big-Branch of the final stage and $\lceil 4 / 3 \rceil = \lceil 4 / 2 \rceil = 2$ .
+
+In all bL-Net variants in which $\beta > 1$ , because we can’t pad in time, we see that the time dimension will get out of sync between the Big-Branch and Little-Branch. Therefore, before merging we need to match the output size of each branch. To do this, we crop the shallower branches in time to match the deepest branch (i.e. the Big-Branch will always have a smaller time dimension due to having more convolutions, so we crop to match it). This is similar to the way the shortcut in ResNet is dealt with in (Saon et al., 2017), and does not introduce edge artifacts when processing longer sequences.
+
+bL-ResNet-22 $( \alpha = 2 , \beta = 3 )$ ) The last of our two-branch models is where $\alpha = 2$ and $\beta = 3$ which is also presented in Table 12. This variant is well-defined in $\alpha$ and $\beta$ up to the first three stages of the network. In the last stage, however, we opt to not branch and instead follow identically the final stage of ResNet-22 with two residual blocks operating at the full input resolution with 512 channels.
+
+bL-PYR-ResNet-22 $( \alpha = 4 , \beta = 1$ ) We additionally present results on a pyramidal structure, in which the first stage of the network operates with four branches, the second with three, the third with two, and the fourth equivalent to the fourth stage of ResNet-22 (and bL-ResNet-22 $( \alpha = 2 , \beta = 3 )$ ). Due to the setting of $\alpha = 4$ , we increased the number of channels in the first stage Big-Branch to have 256 channels (with the three Little-Branches in this stage having 64, 16, and 4 channels), avoiding a single channel on the smallest branch. Note that the middle branches require both resolution upsampling and $1 \times 1$ convolution to match channels. The third stage Big-Little Module operates on two branches and is identical to the analagous stage in bL-ResNet-22 $\alpha = 4 , \beta = 1$ ).

parse/train/HJx81ySKwr/HJx81ySKwr.md

@@ -0,0 +1,271 @@
+# ITERATIVE ENERGY-BASED PROJECTION ON A NORMAL DATA MANIFOLD FOR ANOMALY LOCALIZATION
+
+David Dehaene∗, Oriel Frigo∗, Sébastien Combrexelle, Pierre Eline AnotherBrain, Paris, France {david, oriel, sebastien, pierre}@anotherbrain.ai
+
+# ABSTRACT
+
+Autoencoder reconstructions are widely used for the task of unsupervised anomaly localization. Indeed, an autoencoder trained on normal data is expected to only be able to reconstruct normal features of the data, allowing the segmentation of anomalous pixels in an image via a simple comparison between the image and its autoencoder reconstruction. In practice however, local defects added to a normal image can deteriorate the whole reconstruction, making this segmentation challenging. To tackle the issue, we propose in this paper a new approach for projecting anomalous data on a autoencoder-learned normal data manifold, by using gradient descent on an energy derived from the autoencoder’s loss function. This energy can be augmented with regularization terms that model priors on what constitutes the user-defined optimal projection. By iteratively updating the input of the autoencoder, we bypass the loss of high-frequency information caused by the autoencoder bottleneck. This allows to produce images of higher quality than classic reconstructions. Our method achieves state-of-the-art results on various anomaly localization datasets. It also shows promising results at an inpainting task on the CelebA dataset.
+
+# 1 INTRODUCTION
+
+Automating visual inspection on production lines with artificial intelligence has gained popularity and interest in recent years. Indeed, the analysis of images to segment potential manufacturing defects seems well suited to computer vision algorithms. However these solutions remain data hungry and require knowledge transfer from human to machine via image annotations. Furthermore, the classification in a limited number of user-predefined categories such as non-defective, greasy, scratched and so on, will not generalize well if a previously unseen defect appears. This is even more critical on production lines where a defective product is a rare occurrence. For visual inspection, a better-suited task is unsupervised anomaly detection, in which the segmentation of the defect must be done only via prior knowledge of non-defective samples, constraining the issue to a two-class segmentation problem.
+
+From a statistical point of view, an anomaly may be seen as a distribution outlier, or an observation that deviates so much from other observations as to arouse suspicion that it was generated by a different mechanism (Hawkins, 1980). In this setting, generative models such as Variational AutoEncoders (VAE, Kingma & Welling (2014)), are especially interesting because they are capable to infer possible sampling mechanisms for a given dataset. The original autoencoder (AE) jointly learns an encoder model, that compresses input samples into a low dimensional space, and a decoder, that decompresses the low dimensional samples into the original input space, by minimizing the distance between the input of the encoder and the output of the decoder. The more recent variant, VAE, replaces the deterministic encoder and decoder by stochastic functions, enabling the modeling of the distribution of the dataset samples as well as the generation of new, unseen samples. In both models, the output decompressed sample given an input is often called the reconstruction, and is used as some sort of projection of the input on the support of the normal data distribution, which we will call the normal manifold. In most unsupervised anomaly detection methods based on VAE, models are trained on flawless data and defect detection and localization is then performed using a distance metric between the input sample and its reconstruction (Bergmann et al., 2018; 2019; An & Cho, 2015; Baur et al., 2018; Matsubara et al., 2018).
+
+One fundamental issue in this approach is that the models learn on the normal manifold, hence there is no guarantee of the generalization of their behavior outside this manifold. This is problematic since it is precisely outside the dataset distribution that such methods intend to use the VAE for anomaly localization. Even in the case of a model that always generates credible samples from the dataset distribution, there is no way to ensure that the reconstruction will be connected to the input sample in any useful way. An example illustrating this limitation is given in figure 1, where a VAE trained on regular grid images provides a globally poor reconstruction despite a local perturbation, making the anomaly localization challenging.
+
+In this paper, instead of using the VAE reconstruction, we propose to find a better projection of an input sample on the normal manifold, by optimizing an energy function defined by an autoencoder architecture. Starting at the input sample, we iterate gradient descent steps on the input to converge to an optimum, simultaneously located on the data manifold and closest to the starting input. This method allows us to add prior knowledge about the expected anomalies via regularization terms, which is not possible with the raw VAE reconstruction. We show that such an optimum is better than previously proposed autoencoder reconstructions to localize anomalies on a variety of unsupervised anomaly localization datasets (Bergmann et al., 2019) and present its inpainting capabilities on the CelebA dataset (Liu et al., 2015). We also propose a variant of the standard gradient descent that uses the pixel-wise reconstruction error to speed up the convergence of the energy.
+
+![](images/03e889e1384f99d5b868282f7f8d3e7022b6ec4f46e54b7acbfdf86789ae3a07.jpg)  
+Figure 1: Even though an anomaly is a local perturbation in the image (b), the whole VAEreconstructed image can be disturbed (c). Our gradient descent-based method gives better quality reconstructions (d).
+
+# 2 BACKGROUND
+
+# 2.1 GENERATIVE MODELS
+
+In unsupervised anomaly detection, the only data available during training are samples $\mathbf { x }$ from a non-anomalous dataset $\check { \mathbb { X } } \subset \mathbb { R } ^ { d }$ . In a generative setting, we suppose the existence of a probability function of density $q$ , having its support on all $\mathbb { R } ^ { \bar { d } }$ , from which the dataset was sampled. The generative objective is then to model an estimate of density $q$ , from which we can obtain new samples close to the dataset. Popular generative architectures are Generative Adversarial Networks (GAN, Goodfellow et al. (2014)), that concurrently train a generator $G$ to generate samples from random, low-dimensional noise $\mathbf { z } \ \sim \ p , \ \mathbf { z } \ \in \ \mathbf { \bar { \mathbb { R } } } ^ { l } , \ l \ \ll \ \mathbf { \bar { d } } .$ , and a discriminator $D$ to classify generated samples and dataset samples. This model converges to the equilibrium of the expectation over both real and generated datasets of the binary cross entropy loss of the classifier $m i n _ { G } m a x _ { D }$ [ $\mathbb { E } _ { { \mathbf { x } } \sim q } \left[ \log ( D ( \mathbf { x } ) ) \right] + \mathbb { E } _ { { \mathbf { z } } \sim p } \left[ \log ( 1 - D ( G ( \mathbf { z } ) ) ) \right] ]$ .
+
+Disadvantages of GANs are that they are notoriously difficult to train (Goodfellow, 2017), and they suffer from mode collapse, meaning that they have the tendency to only generate a subset of the original dataset. This can be problematic for anomaly detection, in which we do not want some subset of the normal data to be considered as anomalous (Bergmann et al., 2019). Recent works such as Thanh-Tung et al. (2019) offer simple and attractive explanations for GAN behavior and propose substantial upgrades, however Ravuri & Vinyals (2019) still support the point that GANs have more trouble than other generative models to cover the whole distribution support.
+
+Another generative model is the VAE (Kingma & Welling (2014)), where, similar to a GAN generator, a decoder model tries to approximate the dataset distribution with a simple latent variables prior $p ( \mathbf { z } )$ , with $\mathbf { z } \in \mathbb { R } ^ { l }$ , and conditional distributions output by the decoder $p ( \mathbf { x } | \mathbf { z } )$ . This leads to the estimate $\begin{array} { r } { p ( \mathbf { x } ) = \int p ( \mathbf { x } | \mathbf { z } ) p ( \mathbf { z } ) d z } \end{array}$ , that we would like to optimize using maximum likelihood estimation on the dataset. To render the learning tractable with a stochastic gradient descent (SGD) estimator with reasonable variance, we use importance sampling, introducing density functions $q ( \mathbf { z } | \mathbf { x } )$ output by an encoder network, and Jensen’s inequality to get the variational lower bound :
+
+$$
+\begin{array} { r l } & { \log p ( \mathbf { x } ) = \log \ \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } \mid \mathbf { x } ) } \frac { p ( \mathbf { x } \mid \mathbf { z } ) p ( \mathbf { z } ) } { q ( \mathbf { z } \mid \mathbf { x } ) } } \\ & { \qquad \geq \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } \mid \mathbf { x } ) } \log \ p ( \mathbf { x } \mid \mathbf { z } ) - D _ { \mathrm { K L } } ( q ( \mathbf { z } \mid \mathbf { x } ) \| p ( \mathbf { z } ) ) = - \mathcal { L } ( \mathbf { x } ) } \end{array}
+$$
+
+We will use $\mathcal { L } ( \mathbf { x } )$ as our loss function for training. We define the VAE reconstruction, per analogy with an autoencoder reconstruction, as the deterministic sample $f _ { V A E } ( \mathbf { x } )$ that we obtain by encoding $\mathbf { x }$ , decoding the mean of the encoded distribution $q ( \mathbf { z } | \mathbf { x } )$ , and taking again the mean of the decoded distribution $p ( \mathbf { x } | \mathbf { z } )$ .
+
+VAEs are known to produce blurry reconstructions and generations, but Dai & Wipf (2019) show that a huge enhancement in image quality can be gained by learning the variance of the decoded distribution $p ( \mathbf { x } | \mathbf { z } )$ . This comes at the cost of the distribution of latent variables produced by the encoder $q ( \mathbf { z } )$ being farther away from the prior $p ( \mathbf { z } )$ , so that samples generated by sampling $\mathbf { z } \sim$ $p ( \mathbf { z } ) , \mathbf { x } \sim p ( \mathbf { x } | \mathbf { z } )$ have poorer quality. The authors show that using a second VAE learned on samples from $q ( \mathbf { z } )$ , and sampling from it with ancestral sampling $\mathbf { u } \sim p ( \mathbf { u } ) , \mathbf { z } \sim p ( \mathbf { z } | \mathbf { u } ) , \mathbf { x } \sim p ( \mathbf { x } | \mathbf { z } )$ , allows to recover samples of GAN-like quality. The original autoencoder can be roughly considered as a VAE whose encoded and decoded distributions have infinitely small variances.
+
+# 2.2 ANOMALY DETECTION AND LOCALIZATION
+
+We will consider that an anomaly is a sample with low probability under our estimation of the dataset distribution. The VAE loss, being a lower bound on the density, is a good proxy to classify samples between the anomalous and non-anomalous categories. To this effect, a threshold $T$ can be defined on the loss function, delimiting anomalous samples with ${ \mathcal { L } } ( \mathbf { x } ) \geq T$ and normal samples $\mathcal { L } ( \mathbf { x } ) < T$ . However, according to Matsubara et al. (2018), the regularization term $\mathcal { L } _ { K L } ( \mathbf { x } ) = D _ { \mathrm { K L } } ( q ( \mathbf { z } | \mathbf { x } ) | | p ( \mathbf { z } ) )$ has a negative influence in the computation of anomaly scores. They propose instead an unregularized score $\mathbf { \bar { \mathcal { L } } } _ { r } ( \mathbf { x } ) = - \mathbb { E } _ { \mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } ) }$ log $\bar { p } ( { \bf x } | { \bf z } )$ which is equivalent to the reconstruction term of a standard autoencoder and claim a better anomaly detection.
+
+Going from anomaly detection to anomaly localization, this reconstruction term becomes crucial to most of existing solutions. Indeed, the inability of the model to reconstruct a given part of an image is used as a way to segment the anomaly, using a pixel-wise threshold on the reconstruction error. Actually, this segmentation is very often given by a pixel-wise (An & Cho, 2015; Baur et al., 2018; Matsubara et al., 2018) or patch-wise comparison of the input image, and some generated image, as in Bergmann et al. (2018; 2019), where the structural dissimilarity (DSSIM, Wang et al. (2004)) between the input and its VAE reconstruction is used.
+
+Autoencoder-based methods thus provide a straightforward way of generating an image conditioned on the input image. In the GAN original framework, though, images are generated from random noise $\mathbf { z } \sim p ( \mathbf { z } )$ and are not conditioned by an input. Schlegl et al. (2017) propose with AnoGAN to get the closest generated image to the input using gradient descent on $\mathbf { z }$ for an energy defined by:
+
+$$
+E _ { A n o G A N } = | | \mathbf { x } - G ( \mathbf { z } ) | | _ { 1 } + \lambda \cdot | | f _ { D } ( \mathbf { x } ) - f _ { D } ( G ( \mathbf { z } ) ) | | _ { 1 }
+$$
+
+The first term ensures that the generation $G ( \mathbf { z } )$ is close to the input $\mathbf { x }$ . The second term is based on a distance between features of the input and the generated images, where $f _ { D } ( { \bf x } )$ is the output of an intermediate layer of the discriminator. This term ensures that the generated image stays in the vicinity of the original dataset distribution.
+
+![](images/a40ff0ff9ef4b64b40732d35df0d1beef84532f2f9b25c2bcff5184ca2847666.jpg)  
+Figure 2: Illustration of our method. We perform gradient descent on $E ( \mathbf { x } _ { t } )$ to iteratively correct $\mathbf { x } _ { t }$ .
+
+# 3 PROPOSED METHOD
+
+# 3.1 ADVERSARIAL PROJECTIONS
+
+According to Zimmerer et al. (2018), the loss gradient with respect to $\mathbf { x }$ gives the direction towards normal data samples, and its magnitude could indicate how abnormal a sample is. In their work on anomaly identification, they use the loss gradient as an anomaly score.
+
+Here we propose to use the gradient of the loss to iteratively improve the observed $\mathbf { x }$ . We propose to link this method to the methodology of computing adversarial samples in Szegedy et al. (2014).
+
+After training a VAE on non-anomalous data, we can define a threshold $T$ on the reconstruction loss $\mathcal { L } _ { r }$ as in (Matsubara et al., 2018), such that a small proportion of the most improbable samples are identified as anomalies. We obtain a binary classifier defined by
+
+$$
+A ( \mathbf { x } ) = \left\{ { \begin{array} { l l } { 1 \mathrm { i f } \mathcal { L } _ { r } ( \mathbf { x } ) \geq T } \\ { 0 \mathrm { o t h e r w i s e } } \end{array} } \right.
+$$
+
+Our method consists in computing adversarial samples of this classifier (Szegedy et al., 2014), that is to say, starting from a sample $\mathbf { x } _ { \mathrm { 0 } }$ with $A ( \mathbf { x } _ { 0 } ) = \bar { 1 }$ , iterate gradient descent steps over the input $\mathbf { x }$ , constructing samples $\mathbf { x } _ { 1 } , \ldots . . . \mathbf { x } _ { N }$ , to minimize the energy $E ( \mathbf { x } )$ , defined as
+
+$$
+\begin{array} { r } { E ( \mathbf { x } _ { t } ) = \mathcal { L } _ { r } ( \mathbf { x } _ { t } ) + \lambda \cdot | | \mathbf { x } _ { t } - \mathbf { x } _ { 0 } | | _ { 1 } } \end{array}
+$$
+
+An iteration is done by calculating $\mathbf { x } _ { t + 1 }$ as
+
+$$
+\begin{array} { r } { \mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } - \alpha \cdot \nabla _ { \mathbf { x } } E ( \mathbf { x } _ { t } ) , } \end{array}
+$$
+
+where $\alpha$ is a learning rate parameter, and $\lambda$ is a parameter trading off the inclusion of $\mathbf { x } _ { t }$ in the normal manifold, given by $\mathcal { L } _ { r } ( \mathbf { x } _ { t } )$ , and the proximity between $\mathbf { x } _ { t }$ and the input $\mathbf { x } _ { \mathrm { 0 } }$ , assured by the regularization term $| | \mathbf { x } _ { t } - \mathbf { x } _ { 0 } | | _ { 1 }$ .
+
+# 3.2 REGULARIZATION TERM
+
+We model the anomalous images that we encounter as normal images in which a region or several regions of pixels are altered but the rest of the pixels are left untouched. To recover the best segmentation of the anomalous pixels from an anomalous image $\mathbf { x } _ { a }$ , we want to recover the closest image from the normal manifold $\mathbf { x } _ { g }$ . The term closest has to be understood in the sense that the smallest number of pixels are modified between $\mathbf { x } _ { a }$ and $\mathbf { x } _ { g }$ . In our model, we therefore would like to use the $L ^ { 0 }$ distance as a regularization distance of the energy. Since the $L ^ { 0 }$ distance is not differentiable, we use the $L ^ { 1 }$ distance as an approximation.
+
+# 3.3 OPTIMIZATION IN INPUT SPACE
+
+While in our method the optimization is done in the input space, in the previously mentioned AnoGAN, the search for the optimal reconstruction is done by iterating over $\mathbf { z }$ samples with the
+
+energy defined in equation 2. Following the aforementioned analogy between a GAN generator $G$ and a VAE decoder $D e c$ , a similar approach in the context of a VAE would be to use the energy
+
+$$
+| | \mathbf { x } - D e c ( \mathbf { z } ) | | _ { 1 } - \lambda \cdot \log p ( \mathbf { z } )
+$$
+
+where the $- \log p ( \mathbf { z } )$ term has the same role as AnoGAN’s $| | f _ { D } ( \mathbf { x } ) - f _ { D } ( G ( \mathbf { z } ) ) | | _ { 1 }$ term, to ensure that $D e c ( \mathbf { z } )$ stays within the learned manifold. We chose not to iterate over $\mathbf { z }$ in the latent space for two reasons. First, because as noted in Dai & Wipf (2019) and Hoffman & Johnson (2016), the prior $p ( \mathbf { z } )$ is not always a good proxy for the real image of the distribution in the latent space $q ( \mathbf { z } )$ . Second, because the VAE tends to ignore some details of the original image in its reconstruction, considering that these details are part of the independent pixel noise allowed by the modeling of $p ( \mathbf { x } | \mathbf { z } )$ as a diagonal Gaussian, which causes its infamous blurriness. An optimization in latent space would have to recreate the high frequency structure of the image, whereas iterating over the input image space, and starting the descent on the input image $\mathbf { x } _ { \mathrm { 0 } }$ , allows us to keep that structure and thus to obtain projections of higher quality.
+
+# 3.4 OPTIMIZING GRADIENT DESCENT
+
+We observed that using the Adam optimizer (Kingma & Ba, 2015) is beneficial for the quality of the optimization. Moreover, to speed up the convergence and further preserve the aforementioned high frequency structure of the input, we propose to compute our iterative samples using the pixel-wise reconstruction error of the VAE. To explain the intuition behind this improvement, we will consider the inpainting task. In this setting, as in anomaly localization, a local perturbation is added on top of a normal image. However, in the classic inpainting task, the localization of the perturbation is known beforehand, and we can use the localization mask $\pmb { \Omega }$ to only change the value of the anomalous pixels in the gradient descent:
+
+$$
+\mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } - \alpha \cdot \left( \nabla _ { \mathbf { x } } E ( \mathbf { x } _ { t } ) \odot \Omega \right)
+$$
+
+where $\odot$ is the Hadamard product.
+
+For anomaly localization and blind inpainting, where this information is not available, we compute the pixel-wise reconstruction error which gives a rough estimate of the mask. The term $\nabla _ { \mathbf x } E ( \mathbf x _ { t } )$ is therefore replaced with $\nabla _ { \mathbf x } E ( \mathbf x _ { t } ) \odot ( { \mathbf x _ { t } } - f _ { V A E } ( \mathbf { x } _ { t } ) ) ^ { 2 } .$ ) in equation 5:
+
+$$
+\mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } - \alpha \cdot \left( \nabla _ { \mathbf { x } } E ( \mathbf { x } _ { t } ) \odot ( \mathbf { x } _ { t } - f _ { V A E } ( \mathbf { x } _ { t } ) ) ^ { 2 } \right)
+$$
+
+where $f _ { V A E } ( \mathbf { x } )$ is the standard reconstruction of the VAE. Optimizing the energy this way, a pixel where the reconstruction error is high will update faster, whereas a pixel with good reconstruction will not change easily. This prevents the image to update its pixels where the reconstruction is already good, even with a high learning rate. As can be seen in appendix B, this method converges to the same performance as the method of equation 5, but with fewer iterations. An illustration of our method can be found in figure 2.
+
+# 3.5 STOP CRITERION
+
+A standard stop criterion based on the convergence of the energy can efficiently be used. Using the adversarial setting introduced in section 3.1, we also propose to stop the gradient descent when a certain predefined threshold on the VAE loss is reached. For example, such a threshold can be chosen to be a quantile of the empirical loss distribution computed on the training set.
+
+# 4 EXPERIMENTS
+
+In this section, we evaluate the proposed method for two different applications: anomaly segmentation and image inpainting. Both applications are interesting use cases of our method, where we search to reconstruct partially corrupted images, correcting the anomalies while preserving the uncorrupted image regions.
+
+# 4.1 UNSUPERVISED ANOMALY SEGMENTATION
+
+In order to evaluate the proposed method for the task of anomaly segmentation, we perform experiments with the recently proposed MVTec dataset (Bergmann et al., 2019). This collection of datasets
+
+Table 1: Results for anomaly segmentation on MVTec datasets, expressed in AUROC (Area Under the Receiver Operating Characteristics). Four different baselines are trained on normal samples and are augmented by our proposed gradient based reconstruction (grad) for comparison: A deterministic autoencoder trained with $L ^ { \tilde { 2 } }$ loss $( L ^ { 2 } \mathbf { A E } )$ as in (Bergmann et al., 2019); A deterministic autoencoder trained with DSSIM loss (DSAE) as in (Bergmann et al., 2019); A variational autoencoder (VAE); And a variational autoencoder with a learned decoder variance $\langle \gamma \cdot$ -VAE) as in (Dai & Wipf, 2019). For each result a green or red background denotes respectively an improvement or a decrease in performance compared to the baseline. It can be seen that the proposed gradient-based reconstruction achieves the best segmentation for most datasets, with a mean improvement rate of $9 . 5 2 \%$ over all baselines.
+
+<table><tr><td colspan="2">Category</td><td>L²AE</td><td>L²AE grad</td><td>DSAE</td><td>DSAE grad</td><td>VAE</td><td>VAE grad</td><td>γ-VAE</td><td>γ-VAE grad</td></tr><tr><td rowspan="5">TJ annes</td><td>carpet</td><td>0.539</td><td>0.734</td><td>0.545</td><td>0.774</td><td>0.580</td><td>0.735</td><td>0.648</td><td>0.727</td></tr><tr><td>grid</td><td>0.960</td><td>0.981</td><td>0.960</td><td>0.980</td><td>0.888</td><td>0.961</td><td>0.950</td><td>0.979</td></tr><tr><td>leather</td><td>0.751</td><td>0.921</td><td>0.710</td><td>0.602</td><td>0.834</td><td>0.925</td><td>0.818</td><td>0.897</td></tr><tr><td>tile</td><td>0.476</td><td>0.575</td><td>0.496</td><td>0.626</td><td>0.465</td><td>0.654</td><td>0.491</td><td>0.581</td></tr><tr><td>wood</td><td>0.630</td><td>0.805</td><td>0.641</td><td>0.738</td><td>0.695</td><td>0.838</td><td>0.665</td><td>0.809</td></tr><tr><td rowspan="9">seetqt</td><td>bottle</td><td>0.909</td><td>0.916</td><td>0.933</td><td>0.951</td><td>0.902</td><td>0.922</td><td>0.913</td><td>0.931</td></tr><tr><td>cable</td><td>0.732</td><td>0.864</td><td>0.790</td><td>0.859</td><td>0.828</td><td>0.910</td><td>0.777</td><td>0.880</td></tr><tr><td>capsule</td><td>0.786</td><td>0.952</td><td>0.769</td><td>0.884</td><td>0.862</td><td>0.917</td><td>0.814</td><td>0.917</td></tr><tr><td>hazelnut</td><td>0.976</td><td>0.984</td><td>0.966</td><td>0.966</td><td>0.977</td><td>0.976</td><td>0.977</td><td>0.988</td></tr><tr><td>metalnut</td><td>0.880</td><td>0.899</td><td>0.881</td><td>0.920</td><td>0.881</td><td>0.907</td><td>0.883</td><td>0.914</td></tr><tr><td>pill</td><td>0.885</td><td>0.912</td><td>0.895</td><td>0.927</td><td>0.888</td><td>0.930</td><td>0.897</td><td>0.935</td></tr><tr><td>screw</td><td>0.979</td><td>0.980</td><td>0.983</td><td>0.925</td><td>0.958</td><td>0.945</td><td>0.976</td><td>0.972</td></tr><tr><td>toothbrush</td><td>0.971</td><td>0.983</td><td>0.973</td><td>0.984</td><td>0.971</td><td>0.985</td><td>0.971</td><td>0.983</td></tr><tr><td>transistor</td><td>0.906</td><td>0.921</td><td>0.904</td><td>0.934</td><td>0.894</td><td>0.919</td><td>0.896</td><td>0.931</td></tr><tr><td>zipper</td><td></td><td>0.680</td><td>0.889</td><td>0.828</td><td>0.887</td><td>0.814</td><td>0.869</td><td>0.706</td><td>0.871</td></tr></table>
+
+consists of 15 different categories of objects and textures in the context of industrial inspection, each category containing a number of normal and anomalous samples.
+
+We train our model on normal training samples and test it on both normal and anomalous test samples to evaluate the anomaly segmentation performance.
+
+We perform experiments with three different baseline autoencoders: A “vanilla” variational autoencoder with decoder covariance matrix fixed to identity (Kingma & Welling, 2014), a variational autoencoder with learned decoder variance (Dai & Wipf, 2019), a “vanilla” deterministic autoencoder trained with $L ^ { 2 }$ as reconstruction loss $( L ^ { 2 } \mathrm { A E } )$ and a deterministic autoencoder trained with DSSIM reconstruction loss (DSAE), as proposed by Bergmann et al. (2018). For the sake of a fair comparison, all the autoencoder models are parameterized by convolutional neural networks with the same architecture, latent space dimensionality (set to 100), learning rate (set to 0.0001) and number of epochs (set to 300). The architecture details (layers, paddings, strides) are the same as described in Bergmann et al. (2018) and Bergmann et al. (2019). Similarly to the authors in Bergmann et al. (2019), for the textures datasets, we first subsample the original dataset images to $5 1 2 \times 5 1 2$ and then crop random patches of size $1 2 8 \times 1 2 8$ which are used to train and test the different models. For the object datasets, we directly subsample the original dataset images to $1 2 8 \times 1 2 8$ unlike in Bergmann et al. (2019) who work on $2 5 6 \times 2 5 6$ images, then we perform rotation and translation data augmentations. For all datasets we train on 10000 images.
+
+Anomaly segmentation is then computed by reconstructing the anomalous image and comparing it with the original. We perform the comparison between reconstructed and original with the DSSIM metric as it has been observed in Bergmann et al. (2018) that it provides better anomaly localization than $L ^ { 2 }$ or $L ^ { 1 }$ distances. For the gradient descent, we set the step size $\alpha : = 0 . 5$ , $L ^ { 1 }$ regularization weight $\lambda : = 0 . 0 5$ and the stop criterion is achieved when a sample reconstruction loss is inferior to the minimum reconstruction loss over the training set.
+
+In table 1 we show the AUROC (Area Under the Receiver Operating Characteristics) for different autoencoder methods, with different thresholds applied to the DSSIM anomaly map computed between original and reconstructed images. Note that an AUROC of 1 expresses the best possible segmentation in terms of normal and anomalous pixels. For each autoencoder variant we compare the baseline reconstruction with the proposed gradient-based reconstruction (grad.). As in Bergmann et al. (2019) we observe that an overall best model is hard to identify, however we show that our method increases the AUC values for almost all autoencoder variants. Aggregating the results over all datasets and baselines, we report a mean improvement rate of $9 . 5 2 \%$ , with a median of $4 . 3 3 \%$ , a 25th percentile of $1 . 8 6 \%$ , and a 75th percentile of $1 5 . 8 6 \%$ . The histogram of the improvement rate for all datasets and baselines is provided in appendix F, as well as a short analysis.
+
+In figure 3 we compare our anomaly segmentation with a baseline $L ^ { 2 }$ autoencoder Bergmann et al. (2019) $( L ^ { 2 } \mathbf { A E } )$ for a number of image categories. For all results in figure 3, we set the same threshold to 0.2 to the anomaly detection map given by the DSSIM metric. The visual results in figure 3 highlights an overall improvement of anomaly localization by our proposed iterative reconstruction $\bar { L ^ { 2 } } \mathrm { A } \bar { \mathrm { E } }$ -grad). See appendix C for additional visual results of anomaly segmentation on remaining categories of MVTec dataset, and on remaining baseline models.
+
+![](images/65dae72096b01be8ba39002ab1ce20d45dd35bf9bf93522d3aa43aca83d05f6c.jpg)  
+Figure 3: First row: Normal samples of hazelnut, grid, cable, wood, carpet and bottle categories in MVTec dataset; Second row: anomalous samples from the aforementioned dataset categories; Third row: Anomaly segmentation with baseline $\bar { L } ^ { 2 }$ autoencoder (Bergmann et al., 2019); Fourth row: our proposed anomaly segmentation with $L ^ { 2 }$ autoencoder augmented with gradient-based iterative reconstruction. Ground truth is represented by red contour, and each estimated segmentation by a green overlay. It can be seen that anomaly segmentation is refined by our proposed method, with a tendency of detecting less false positives.
+
+# 4.2 INPAINTING
+
+Image inpainting is a well known image reconstruction problem which consists of reconstructing a corrupted or missing part of an image, where the region to be reconstructed is usually given by a known mask. Many different approaches for inpainting have been proposed in the literature, such as anisotropic diffusion (Bertalmio et al., 2000), patch matching (Criminisi et al., 2004), context autoencoders (Pathak et al., 2016) and conditional variational autoencoders (Ivanov et al., 2019).
+
+If we consider that the region to be reconstructed is not known beforehand, the problem is sometimes called blind inpainting (Altinel et al., 2018), and the corrupted part can be seen as an anomaly to be corrected.
+
+![](images/3c525dc352b059d364d4bd4eb4992154d606ff497f94c348f4d97f8db18a4df2.jpg)  
+Figure 4: Inpainting experiment performed on CelebA dataset, where the test face images are masked with uniform noise. The baseline VAE reconstruction is disturbed by the noise mask, providing a poor inpainting. The proposed gradient-based VAE provides a more convincing inpainting by an iterative process.
+
+We performed experiments with image inpainting on the CelebA dataset (Liu et al., 2015), which consists of celebrity faces. In figure 4 we compare the inpainting results obtained with a baseline VAE with learned variance $\left( \gamma { \mathrm { - V A E } } \right)$ and Resnet architecture, as described by Dai & Wipf (2019), with the same VAE model, augmented by our proposed gradient-based iterative reconstruction. Note that for the regular inpainting task, gradients are multiplied by the inpainting mask at each iteration (equation 7), while for the blind inpainting task, the mask is unknown. See appendix D for a comparison with a recent method based on variational autoencoders, proposed by Ivanov et al. (2019).
+
+# 5 RELATED WORK
+
+Baur et al. (2018) have used autoencoder reconstructions to localize anomalies in MRI scans, and have compared several variants using diverse per-pixel distances as well as perceptual metrics derived from a GAN-like architecture. Bergmann et al. (2018) use the structural similarity metric (Wang et al., 2004) to compare the original image and its reconstruction to achieve better anomaly localization, and also presents the SSIM autoencoder, which is trained directly with this metric.
+
+Zimmerer et al. (2018) use the derivative of the VAE loss function with respect to the input, called the score. The amplitude of the score is supposed to indicate how abnormal a pixel is. While we agree that the gradient of the loss is an indication of an anomaly, we think that we have to integrate this gradient over the path from the input to the normal manifold to obtain meaningful information. We compare our results to score-based results for anomaly localization in appendix A.
+
+The work that is the most related to ours is AnoGAN (Schlegl et al., 2017). We have mentioned above the differences between the two approaches, which, apart from the change in underlying architectures, boil down to the ability in our method to update directly the input image instead of searching for the optimal latent code. This enables the method to converge faster and above all to keep higher-frequency structures of the input, which would have been deteriorated if it were passed through the AE bottleneck. Bergmann et al. (2019) compare standard AE reconstructions techniques to AnoGAN, and observes that AnoGAN’s performances on anomaly localizations tasks are poorer than AE’s due to the mode collapse tendency of GAN architectures. Interestingly, updates on AnoGAN such as fast AnoGAN (Schlegl et al., 2019) or AnoVAEGAN (Baur et al., 2018) replaced the gradient descent search of the optimal z with a learned encoder model, yielding an approach very similar to the standard VAE reconstruction-based approaches, but with a reconstruction loss learned by a discriminator, which is still prone to mode collapse (Thanh-Tung et al., 2019).
+
+# 6 CONCLUSION
+
+In this paper, we proposed a novel method for unsupervised anomaly localization, using gradient descent of an energy defined by an autoencoder reconstruction loss. Starting from a sample under test, we iteratively update this sample to reduce its autoencoder reconstruction error. This method offers a way to incorporate human priors into what is the optimal projection of an out-of-distribution sample into the normal data manifold. In particular, we use the pixel-wise reconstruction error to modulate the gradient descent, which gives impressive anomaly localization results in only a few iterations. Using gradient descent in the input data space, starting from the input sample, enables us to overcome the autoencoder tendency to provide blurry reconstructions and keep normal high frequency structures. This significantly reduces the number of pixels that could be wrongly classified as defects when the autoencoder fails to reconstruct high frequencies. We showed that this method, which can easily be added to any previously trained autoencoder architecture, gives state-of-the-art results on a variety of unsupervised anomaly localization datasets, as well as qualitative reconstructions on an inpainting task. Future work can focus on replacing the $L ^ { 1 }$ -based regularization term with a Bayesian prior modeling common types of anomalies, and on further improving the speed of the gradient descent.
+
+# REFERENCES
+
+Fazil Altinel, Mete Ozay, and Takayuki Okatani. Deep structured energy-based image inpainting. In 2018 24th International Conference on Pattern Recognition (ICPR), pp. 423–428, Aug 2018. doi: 10.1109/ICPR.2018.8546025.
+
+Jinwon An and Sungzoon Cho. Variational autoencoder based anomaly detection using reconstruction probability. Technical report, SNU Data Mining Center, 2015.
+
+Christoph Baur, Benedikt Wiestler, Shadi Albarqouni, and Nassir Navab. Deep autoencoding models for unsupervised anomaly segmentation in brain MR images. CoRR, abs/1804.04488, 2018.
+
+Paul Bergmann, Sindy Löwe, Michael Fauser, David Sattlegger, and Carsten Steger. Improving unsupervised defect segmentation by applying structural similarity to autoencoders. CoRR, abs/1807.02011, 2018.
+
+Paul Bergmann, Michael Fauser, David Sattlegger, and Carsten Steger. Mvtec ad — a comprehensive real-world dataset for unsupervised anomaly detection. CVPR, 2019.
+
+Marcelo Bertalmio, Guillermo Sapiro, Vincent Caselles, and Coloma Ballester. Image inpainting. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH $\mathbf { \dot { \theta } } _ { 0 0 }$ , pp. 417–424, New York, NY, USA, 2000. ACM Press/Addison-Wesley Publishing Co. ISBN 1-58113-208-5. doi: 10.1145/344779.344972.
+
+Antonio Criminisi, Patrick Pérez, and Kentaro Toyama. Region filling and object removal by exemplar-based image inpainting. Trans. Img. Proc., 13(9):1200–1212, September 2004. ISSN 1057-7149. doi: 10.1109/TIP.2004.833105.
+
+Bin Dai and David P. Wipf. Diagnosing and enhancing VAE models. CoRR, abs/1903.05789, 2019.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.
+
+Ian J. Goodfellow. NIPS 2016 tutorial: Generative adversarial networks. CoRR, abs/1701.00160, 2017.
+
+Douglas M Hawkins. Identification of outliers. Monographs on applied probability and statistics. Chapman and Hall, London [u.a.], 1980. ISBN 041221900X.
+
+Matthew D. Hoffman and Matthew J. Johnson. Elbo surgery: yet another way to carve up the variational evidence lower bound. In NIPS 2016 Workshop on Advances in Approximate Bayesian Inference, 2016.
+
+Oleg Ivanov, Michael Figurnov, and Dmitry Vetrov. Variational autoencoder with arbitrary conditioning. In International Conference on Learning Representations, 2019.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun (eds.), 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015.
+
+Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In Yoshua Bengio and Yann LeCun (eds.), 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014.
+
+Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of International Conference on Computer Vision (ICCV), December 2015.
+
+Takashi Matsubara, Ryosuke Tachibana, and Kuniaki Uehara. Anomaly machine component detection by deep generative model with unregularized score. CoRR, abs/1807.05800, 2018.
+
+Deepak Pathak, Philipp Krähenbühl, Jeff Donahue, Trevor Darrell, and Alexei Efros. Context encoders: Feature learning by inpainting. In Computer Vision and Pattern Recognition (CVPR), 2016.
+
+Suman V. Ravuri and Oriol Vinyals. Classification accuracy score for conditional generative models. CoRR, abs/1905.10887, 2019.
+
+Thomas Schlegl, Philipp Seeböck, Sebastian M Waldstein, Ursula Schmidt-Erfurth, and Georg Langs. Unsupervised anomaly detection with generative adversarial networks to guide marker discovery. In International Conference on Information Processing in Medical Imaging, pp. 146– 157. Springer, 2017.
+
+Thomas Schlegl, Philipp Seeböck, Sebastian M. Waldstein, Georg Langs, and Ursula SchmidtErfurth. f-anogan: Fast unsupervised anomaly detection with generative adversarial networks. Medical Image Analysis, 54:30 – 44, 2019. ISSN 1361-8415. doi: https://doi.org/10.1016/j. media.2019.01.010.
+
+Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian J. Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In Yoshua Bengio and Yann LeCun (eds.), 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014.
+
+Hoang Thanh-Tung, Truyen Tran, and Svetha Venkatesh. Improving generalization and stability of generative adversarial networks. CoRR, abs/1902.03984, 2019.
+
+Zhou Wang, Alan Bovik, Hamid Sheikh, and Eero Simoncelli. Image quality assessment: From error visibility to structural similarity. Image Processing, IEEE Transactions on, 13:600 – 612, 05 2004. doi: 10.1109/TIP.2003.819861.
+
+David Zimmerer, Fabian Isensee, Jens Petersen, Simon Kohl A. A., and Klaus H. Maier-Hein. Unsupervised anomaly localization using variational auto-encoders. CoRR, abs/1907.02796, 2019.
+
+# A COMPARISON WITH ZIMMERER ET AL. (2019)
+
+Table 2: Complementary results for anomaly segmentation on MVTec datasets, expressed in AUROC for different pixel-wise scores derived from a baseline VAE: from left to right, squared error reconstruction $| | { \bf x } - f _ { V A E } ( { \bf x } ) | | ^ { 2 }$ (denoted $\mathcal { L } _ { r } ( \mathbf { x } ) )$ , gradient of the loss $| \nabla _ { \mathbf x } \mathcal L ( \mathbf x ) |$ , combination of both, gradient of the KL divergence $| D _ { \mathrm { K L } } ( q ( \mathbf { z } | \mathbf { x } ) \| p ( \mathbf { z } ) )$ (denoted $| \nabla _ { \mathbf { x } } \mathcal { L } _ { K L } ( \mathbf { x } ) | )$ as well as combination of KL derivative and error reconstruction as suggested in Zimmerer et al. (2018; 2019).   
+
+<table><tr><td></td><td>Category</td><td>Lr(x)</td><td>VxL(x)</td><td>IVxL(x)) OLr(x)</td><td>IVxLKL(x)l</td><td>IVxLKL(x)) Lr(x)</td><td>VAE-grad</td></tr><tr><td rowspan="5">TJssnets</td><td>carpet</td><td>0.537</td><td>0.580</td><td>0.566</td><td>0.553</td><td>0.555</td><td>0.735</td></tr><tr><td>grid</td><td>0.823</td><td>0.635</td><td>0.812</td><td>0.507</td><td>0.790</td><td>0.961</td></tr><tr><td>leather</td><td>0.783</td><td>0.650</td><td>0.792</td><td>0.627</td><td>0.791</td><td>0.925</td></tr><tr><td>tile</td><td>0.547</td><td>0.606</td><td>0.581</td><td>0.623</td><td>0.588</td><td>0.654</td></tr><tr><td>wood</td><td>0.686</td><td>0.691</td><td>0.726</td><td>0.643</td><td>0.713</td><td>0.838</td></tr><tr><td rowspan="10">seetqt</td><td>bottle</td><td>0.831</td><td>0.762</td><td>0.832</td><td>0.629</td><td>0.830</td><td>0.922</td></tr><tr><td>cable</td><td>0.831</td><td>0.796</td><td>0.846</td><td>0.674</td><td>0.841</td><td>0.910</td></tr><tr><td>capsule</td><td>0.765</td><td>0.754</td><td>0.772</td><td>0.642</td><td>0.795</td><td>0.917.</td></tr><tr><td>hazelnut</td><td>0.907</td><td>0.831</td><td>0.908</td><td>0.468</td><td>0.885</td><td>0.976</td></tr><tr><td>metalnut</td><td>0.833</td><td>0.831</td><td>0.870</td><td>0.710</td><td>0.834</td><td>0.907</td></tr><tr><td>pill</td><td>0.869</td><td>0.833</td><td>0.872</td><td>0.480</td><td>0.826</td><td>0.930</td></tr><tr><td>screw</td><td>0.851</td><td>0.726</td><td>0.842</td><td>0.412</td><td>0.795</td><td>0.945</td></tr><tr><td>toothbrush</td><td>0.942</td><td>0.798</td><td>0.943</td><td>0.619</td><td>0.939</td><td>0.985</td></tr><tr><td>transistor</td><td>0.788</td><td>0.843</td><td>0.834</td><td>0.801</td><td>0.836</td><td>0.919</td></tr><tr><td>zipper</td><td>0.725</td><td>0.674</td><td>0.729</td><td>0.562</td><td>0.727</td><td>0.869</td></tr></table>
+
+Zimmerer et al. (2019) proposed to perform anomaly localization using different scores derived from the gradient of the VAE loss. In particular, it has been shown that the product of the VAE reconstruction error with the gradient of the KL divergence was very informative for medical images. In table 2 we compare the pixel-wise anomaly detection AUROC of these different scores with our method. For all experiments, we use the same “vanilla” VAE as described in section 4.1.
+
+It can be seen that other VAE-based methods using a single evaluation of the gradient are constantly outperformed by our method.
+
+# B CONVERGENCE SPEED
+
+![](images/345cea3215d5de240941e391b07b8e8c6ecd6b2a51cb52dde3287447ce2a70b3.jpg)  
+Figure 5: Evolution of pixel-wise anomaly detection AUROC performance.
+
+In figure 5 we compare the number of iterations needed to reach convergence with our two proposals for gradient descent: Standard update as in equation 5 and Tuned update using a gradient mask computed with the VAE reconstruction error, as in equation 8. The model is a VAE with learned decoder variance (Dai & Wipf, 2019), trained on the Grid dataset (Bergmann et al., 2019). We compute the mean pixel-wise anomaly detection AUROC after each iteration on the test set.
+
+We can see that the tuned method converges to the same performance as the standard method, with far fewer iterations.
+
+![](images/2cebada9b7197e00c2fab69f1850ebdeacfeefa2a889bd15909f9a0d95c80f61.jpg)  
+Figure 6: From left to right: Normal; Anomalous; Anomaly segmentation with baseline $L ^ { 2 }$ autoencoder (Bergmann et al., 2019); Our proposed anomaly segmentation with $L ^ { 2 }$ autoencoder augmented with gradient-based iterative reconstruction.
+
+![](images/0a930ed3a0d4bf482f1bb30b17161e0bf7c1b209bebf56b1660f760c38e4db58.jpg)  
+Figure 7: Illustration of anomaly localization comparison over four baselines ( $L ^ { 2 } { \mathrm { A E } }$ , DSAE, VAE, $\gamma$ -VAE). Ground truth is represented by red contour, and each estimated segmentation by a green overlay. It can be seen that anomaly segmentation is overall improved when different baselines are augmented by our proposed gradient descent.
+
+# D INPAINTING COMPARISON
+
+![](images/ffc49efc49762ecb3f4c500413e7267bb1d6f339702e0013a7cde29f0684c8c5.jpg)  
+Figure 8: Inpainting comparison. Each batch is made of four rows, from top to bottom: Masked input image; VAE with arbitrary conditioning (VAEAC, Ivanov et al. (2019)); Ours; Ground truth. The quality of the reconstructions is comparable, even though our VAE is trained without any assumptions over the mask’s properties.
+
+![](images/2a43dfab4dcfee18aabf106a89fa776779a096451340a5174bf902c8a544094a.jpg)  
+Figure 9: Principle of the energy optimization to project anomalous sample on the normal manifold
+
+Figure 9 illustrates our method principle. We start with a defective input $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ whose reconstruction $\scriptstyle { \hat { \mathbf { x } } } _ { 0 }$ does not necessarily lie on the normal data manifold. As the optimization process carries on, the optimized sample $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ and its reconstruction look more similar and get closer to the manifold. The regularization term of the energy function makes sure that the optimized sample stays close to the original sample.
+
+![](images/af15f62eeb6a6ad21ae17c81d2036303dc40784f6c062bc7493c7a6a7a566a04.jpg)  
+Figure 10: Distribution of the improvement rate over all presented baselines and all datasets in MVTec AD.
+
+Figure 10 shows the distribution of the AUC improvement rate over all presented baselines and all datasets in MVTec AD using our gradient-based projection method.
+
+$$
+{ \mathrm { i m p r o v e m e n t r a t e } } = { \frac { A U C _ { g r a d } - A U C _ { b a s e } } { A U C _ { b a s e } } }
+$$
+
+• $8 . 3 \%$ of data points are under the 0 value delimiting an increase or decrease in AUC due to our method, and $9 1 . 7 \%$ data points are over this value. Our method increases the AUC in a vast majority of cases.
+
+• The median is at $4 . 3 3 \%$ , the 25th percentile at $1 . 8 6 \%$ , and the 75th percentile at $1 5 . 8 6 \%$ .

parse/train/HJx81ySKwr/HJx81ySKwr_content_list.json

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+    {
+        "type": "text",
+        "text": "ITERATIVE ENERGY-BASED PROJECTION ON A NORMAL DATA MANIFOLD FOR ANOMALY LOCALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            101,
+            820,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "David Dehaene∗, Oriel Frigo∗, Sébastien Combrexelle, Pierre Eline AnotherBrain, Paris, France {david, oriel, sebastien, pierre}@anotherbrain.ai ",
+        "bbox": [
+            184,
+            169,
+            661,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Autoencoder reconstructions are widely used for the task of unsupervised anomaly localization. Indeed, an autoencoder trained on normal data is expected to only be able to reconstruct normal features of the data, allowing the segmentation of anomalous pixels in an image via a simple comparison between the image and its autoencoder reconstruction. In practice however, local defects added to a normal image can deteriorate the whole reconstruction, making this segmentation challenging. To tackle the issue, we propose in this paper a new approach for projecting anomalous data on a autoencoder-learned normal data manifold, by using gradient descent on an energy derived from the autoencoder’s loss function. This energy can be augmented with regularization terms that model priors on what constitutes the user-defined optimal projection. By iteratively updating the input of the autoencoder, we bypass the loss of high-frequency information caused by the autoencoder bottleneck. This allows to produce images of higher quality than classic reconstructions. Our method achieves state-of-the-art results on various anomaly localization datasets. It also shows promising results at an inpainting task on the CelebA dataset. ",
+        "bbox": [
+            233,
+            281,
+            764,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            529,
+            334,
+            545
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Automating visual inspection on production lines with artificial intelligence has gained popularity and interest in recent years. Indeed, the analysis of images to segment potential manufacturing defects seems well suited to computer vision algorithms. However these solutions remain data hungry and require knowledge transfer from human to machine via image annotations. Furthermore, the classification in a limited number of user-predefined categories such as non-defective, greasy, scratched and so on, will not generalize well if a previously unseen defect appears. This is even more critical on production lines where a defective product is a rare occurrence. For visual inspection, a better-suited task is unsupervised anomaly detection, in which the segmentation of the defect must be done only via prior knowledge of non-defective samples, constraining the issue to a two-class segmentation problem. ",
+        "bbox": [
+            174,
+            560,
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+            699
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "From a statistical point of view, an anomaly may be seen as a distribution outlier, or an observation that deviates so much from other observations as to arouse suspicion that it was generated by a different mechanism (Hawkins, 1980). In this setting, generative models such as Variational AutoEncoders (VAE, Kingma & Welling (2014)), are especially interesting because they are capable to infer possible sampling mechanisms for a given dataset. The original autoencoder (AE) jointly learns an encoder model, that compresses input samples into a low dimensional space, and a decoder, that decompresses the low dimensional samples into the original input space, by minimizing the distance between the input of the encoder and the output of the decoder. The more recent variant, VAE, replaces the deterministic encoder and decoder by stochastic functions, enabling the modeling of the distribution of the dataset samples as well as the generation of new, unseen samples. In both models, the output decompressed sample given an input is often called the reconstruction, and is used as some sort of projection of the input on the support of the normal data distribution, which we will call the normal manifold. In most unsupervised anomaly detection methods based on VAE, models are trained on flawless data and defect detection and localization is then performed using a distance metric between the input sample and its reconstruction (Bergmann et al., 2018; 2019; An & Cho, 2015; Baur et al., 2018; Matsubara et al., 2018). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One fundamental issue in this approach is that the models learn on the normal manifold, hence there is no guarantee of the generalization of their behavior outside this manifold. This is problematic since it is precisely outside the dataset distribution that such methods intend to use the VAE for anomaly localization. Even in the case of a model that always generates credible samples from the dataset distribution, there is no way to ensure that the reconstruction will be connected to the input sample in any useful way. An example illustrating this limitation is given in figure 1, where a VAE trained on regular grid images provides a globally poor reconstruction despite a local perturbation, making the anomaly localization challenging. ",
+        "bbox": [
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+            138,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, instead of using the VAE reconstruction, we propose to find a better projection of an input sample on the normal manifold, by optimizing an energy function defined by an autoencoder architecture. Starting at the input sample, we iterate gradient descent steps on the input to converge to an optimum, simultaneously located on the data manifold and closest to the starting input. This method allows us to add prior knowledge about the expected anomalies via regularization terms, which is not possible with the raw VAE reconstruction. We show that such an optimum is better than previously proposed autoencoder reconstructions to localize anomalies on a variety of unsupervised anomaly localization datasets (Bergmann et al., 2019) and present its inpainting capabilities on the CelebA dataset (Liu et al., 2015). We also propose a variant of the standard gradient descent that uses the pixel-wise reconstruction error to speed up the convergence of the energy. ",
+        "bbox": [
+            174,
+            257,
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+            397
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/03e889e1384f99d5b868282f7f8d3e7022b6ec4f46e54b7acbfdf86789ae3a07.jpg",
+        "image_caption": [
+            "Figure 1: Even though an anomaly is a local perturbation in the image (b), the whole VAEreconstructed image can be disturbed (c). Our gradient descent-based method gives better quality reconstructions (d). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+            636
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 GENERATIVE MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In unsupervised anomaly detection, the only data available during training are samples $\\mathbf { x }$ from a non-anomalous dataset $\\check { \\mathbb { X } } \\subset \\mathbb { R } ^ { d }$ . In a generative setting, we suppose the existence of a probability function of density $q$ , having its support on all $\\mathbb { R } ^ { \\bar { d } }$ , from which the dataset was sampled. The generative objective is then to model an estimate of density $q$ , from which we can obtain new samples close to the dataset. Popular generative architectures are Generative Adversarial Networks (GAN, Goodfellow et al. (2014)), that concurrently train a generator $G$ to generate samples from random, low-dimensional noise $\\mathbf { z } \\ \\sim \\ p , \\ \\mathbf { z } \\ \\in \\ \\mathbf { \\bar { \\mathbb { R } } } ^ { l } , \\ l \\ \\ll \\ \\mathbf { \\bar { d } } .$ , and a discriminator $D$ to classify generated samples and dataset samples. This model converges to the equilibrium of the expectation over both real and generated datasets of the binary cross entropy loss of the classifier $m i n _ { G } m a x _ { D }$ [ $\\mathbb { E } _ { { \\mathbf { x } } \\sim q } \\left[ \\log ( D ( \\mathbf { x } ) ) \\right] + \\mathbb { E } _ { { \\mathbf { z } } \\sim p } \\left[ \\log ( 1 - D ( G ( \\mathbf { z } ) ) ) \\right] ]$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Disadvantages of GANs are that they are notoriously difficult to train (Goodfellow, 2017), and they suffer from mode collapse, meaning that they have the tendency to only generate a subset of the original dataset. This can be problematic for anomaly detection, in which we do not want some subset of the normal data to be considered as anomalous (Bergmann et al., 2019). Recent works such as Thanh-Tung et al. (2019) offer simple and attractive explanations for GAN behavior and propose substantial upgrades, however Ravuri & Vinyals (2019) still support the point that GANs have more trouble than other generative models to cover the whole distribution support. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another generative model is the VAE (Kingma & Welling (2014)), where, similar to a GAN generator, a decoder model tries to approximate the dataset distribution with a simple latent variables prior $p ( \\mathbf { z } )$ , with $\\mathbf { z } \\in \\mathbb { R } ^ { l }$ , and conditional distributions output by the decoder $p ( \\mathbf { x } | \\mathbf { z } )$ . This leads to the estimate $\\begin{array} { r } { p ( \\mathbf { x } ) = \\int p ( \\mathbf { x } | \\mathbf { z } ) p ( \\mathbf { z } ) d z } \\end{array}$ , that we would like to optimize using maximum likelihood estimation on the dataset. To render the learning tractable with a stochastic gradient descent (SGD) estimator with reasonable variance, we use importance sampling, introducing density functions $q ( \\mathbf { z } | \\mathbf { x } )$ output by an encoder network, and Jensen’s inequality to get the variational lower bound : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b4eca21592992aaa2359b7d1ed03f84b5d6efc2ffa3706c3d3dbff463e11c1d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log p ( \\mathbf { x } ) = \\log \\ \\mathbb { E } _ { \\mathbf { z } \\sim q ( \\mathbf { z } \\mid \\mathbf { x } ) } \\frac { p ( \\mathbf { x } \\mid \\mathbf { z } ) p ( \\mathbf { z } ) } { q ( \\mathbf { z } \\mid \\mathbf { x } ) } } \\\\ & { \\qquad \\geq \\mathbb { E } _ { \\mathbf { z } \\sim q ( \\mathbf { z } \\mid \\mathbf { x } ) } \\log \\ p ( \\mathbf { x } \\mid \\mathbf { z } ) - D _ { \\mathrm { K L } } ( q ( \\mathbf { z } \\mid \\mathbf { x } ) \\| p ( \\mathbf { z } ) ) = - \\mathcal { L } ( \\mathbf { x } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "We will use $\\mathcal { L } ( \\mathbf { x } )$ as our loss function for training. We define the VAE reconstruction, per analogy with an autoencoder reconstruction, as the deterministic sample $f _ { V A E } ( \\mathbf { x } )$ that we obtain by encoding $\\mathbf { x }$ , decoding the mean of the encoded distribution $q ( \\mathbf { z } | \\mathbf { x } )$ , and taking again the mean of the decoded distribution $p ( \\mathbf { x } | \\mathbf { z } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "VAEs are known to produce blurry reconstructions and generations, but Dai & Wipf (2019) show that a huge enhancement in image quality can be gained by learning the variance of the decoded distribution $p ( \\mathbf { x } | \\mathbf { z } )$ . This comes at the cost of the distribution of latent variables produced by the encoder $q ( \\mathbf { z } )$ being farther away from the prior $p ( \\mathbf { z } )$ , so that samples generated by sampling $\\mathbf { z } \\sim$ $p ( \\mathbf { z } ) , \\mathbf { x } \\sim p ( \\mathbf { x } | \\mathbf { z } )$ have poorer quality. The authors show that using a second VAE learned on samples from $q ( \\mathbf { z } )$ , and sampling from it with ancestral sampling $\\mathbf { u } \\sim p ( \\mathbf { u } ) , \\mathbf { z } \\sim p ( \\mathbf { z } | \\mathbf { u } ) , \\mathbf { x } \\sim p ( \\mathbf { x } | \\mathbf { z } )$ , allows to recover samples of GAN-like quality. The original autoencoder can be roughly considered as a VAE whose encoded and decoded distributions have infinitely small variances. ",
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+    {
+        "type": "text",
+        "text": "2.2 ANOMALY DETECTION AND LOCALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We will consider that an anomaly is a sample with low probability under our estimation of the dataset distribution. The VAE loss, being a lower bound on the density, is a good proxy to classify samples between the anomalous and non-anomalous categories. To this effect, a threshold $T$ can be defined on the loss function, delimiting anomalous samples with ${ \\mathcal { L } } ( \\mathbf { x } ) \\geq T$ and normal samples $\\mathcal { L } ( \\mathbf { x } ) < T$ . However, according to Matsubara et al. (2018), the regularization term $\\mathcal { L } _ { K L } ( \\mathbf { x } ) = D _ { \\mathrm { K L } } ( q ( \\mathbf { z } | \\mathbf { x } ) | | p ( \\mathbf { z } ) )$ has a negative influence in the computation of anomaly scores. They propose instead an unregularized score $\\mathbf { \\bar { \\mathcal { L } } } _ { r } ( \\mathbf { x } ) = - \\mathbb { E } _ { \\mathbf { z } \\sim q ( \\mathbf { z } | \\mathbf { x } ) }$ log $\\bar { p } ( { \\bf x } | { \\bf z } )$ which is equivalent to the reconstruction term of a standard autoencoder and claim a better anomaly detection. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Going from anomaly detection to anomaly localization, this reconstruction term becomes crucial to most of existing solutions. Indeed, the inability of the model to reconstruct a given part of an image is used as a way to segment the anomaly, using a pixel-wise threshold on the reconstruction error. Actually, this segmentation is very often given by a pixel-wise (An & Cho, 2015; Baur et al., 2018; Matsubara et al., 2018) or patch-wise comparison of the input image, and some generated image, as in Bergmann et al. (2018; 2019), where the structural dissimilarity (DSSIM, Wang et al. (2004)) between the input and its VAE reconstruction is used. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Autoencoder-based methods thus provide a straightforward way of generating an image conditioned on the input image. In the GAN original framework, though, images are generated from random noise $\\mathbf { z } \\sim p ( \\mathbf { z } )$ and are not conditioned by an input. Schlegl et al. (2017) propose with AnoGAN to get the closest generated image to the input using gradient descent on $\\mathbf { z }$ for an energy defined by: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2cef368449a906ba7a47ee14692a9cc36d2c8785c2ccbb17b7a10b7436c14cf1.jpg",
+        "text": "$$\nE _ { A n o G A N } = | | \\mathbf { x } - G ( \\mathbf { z } ) | | _ { 1 } + \\lambda \\cdot | | f _ { D } ( \\mathbf { x } ) - f _ { D } ( G ( \\mathbf { z } ) ) | | _ { 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The first term ensures that the generation $G ( \\mathbf { z } )$ is close to the input $\\mathbf { x }$ . The second term is based on a distance between features of the input and the generated images, where $f _ { D } ( { \\bf x } )$ is the output of an intermediate layer of the discriminator. This term ensures that the generated image stays in the vicinity of the original dataset distribution. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a40ff0ff9ef4b64b40732d35df0d1beef84532f2f9b25c2bcff5184ca2847666.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of our method. We perform gradient descent on $E ( \\mathbf { x } _ { t } )$ to iteratively correct $\\mathbf { x } _ { t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 ADVERSARIAL PROJECTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "According to Zimmerer et al. (2018), the loss gradient with respect to $\\mathbf { x }$ gives the direction towards normal data samples, and its magnitude could indicate how abnormal a sample is. In their work on anomaly identification, they use the loss gradient as an anomaly score. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here we propose to use the gradient of the loss to iteratively improve the observed $\\mathbf { x }$ . We propose to link this method to the methodology of computing adversarial samples in Szegedy et al. (2014). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "After training a VAE on non-anomalous data, we can define a threshold $T$ on the reconstruction loss $\\mathcal { L } _ { r }$ as in (Matsubara et al., 2018), such that a small proportion of the most improbable samples are identified as anomalies. We obtain a binary classifier defined by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/a531f29019642757f994e7aa898c3c8061e00933d98c93d5b91f6ebd5032bea7.jpg",
+        "text": "$$\nA ( \\mathbf { x } ) = \\left\\{ { \\begin{array} { l l } { 1 \\mathrm { i f } \\mathcal { L } _ { r } ( \\mathbf { x } ) \\geq T } \\\\ { 0 \\mathrm { o t h e r w i s e } } \\end{array} } \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our method consists in computing adversarial samples of this classifier (Szegedy et al., 2014), that is to say, starting from a sample $\\mathbf { x } _ { \\mathrm { 0 } }$ with $A ( \\mathbf { x } _ { 0 } ) = \\bar { 1 }$ , iterate gradient descent steps over the input $\\mathbf { x }$ , constructing samples $\\mathbf { x } _ { 1 } , \\ldots . . . \\mathbf { x } _ { N }$ , to minimize the energy $E ( \\mathbf { x } )$ , defined as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d6c1c1f32e9cf8aa30fffa53409b26b381a130bfc27957c14ade92eb05865fc.jpg",
+        "text": "$$\n\\begin{array} { r } { E ( \\mathbf { x } _ { t } ) = \\mathcal { L } _ { r } ( \\mathbf { x } _ { t } ) + \\lambda \\cdot | | \\mathbf { x } _ { t } - \\mathbf { x } _ { 0 } | | _ { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "An iteration is done by calculating $\\mathbf { x } _ { t + 1 }$ as ",
+        "bbox": [
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+            613,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58510f5a009e1ed2a0b4e32f6ecede5b14e904e0d546a4b5dab7e12604e40aa7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - \\alpha \\cdot \\nabla _ { \\mathbf { x } } E ( \\mathbf { x } _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            635,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha$ is a learning rate parameter, and $\\lambda$ is a parameter trading off the inclusion of $\\mathbf { x } _ { t }$ in the normal manifold, given by $\\mathcal { L } _ { r } ( \\mathbf { x } _ { t } )$ , and the proximity between $\\mathbf { x } _ { t }$ and the input $\\mathbf { x } _ { \\mathrm { 0 } }$ , assured by the regularization term $| | \\mathbf { x } _ { t } - \\mathbf { x } _ { 0 } | | _ { 1 }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 REGULARIZATION TERM ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We model the anomalous images that we encounter as normal images in which a region or several regions of pixels are altered but the rest of the pixels are left untouched. To recover the best segmentation of the anomalous pixels from an anomalous image $\\mathbf { x } _ { a }$ , we want to recover the closest image from the normal manifold $\\mathbf { x } _ { g }$ . The term closest has to be understood in the sense that the smallest number of pixels are modified between $\\mathbf { x } _ { a }$ and $\\mathbf { x } _ { g }$ . In our model, we therefore would like to use the $L ^ { 0 }$ distance as a regularization distance of the energy. Since the $L ^ { 0 }$ distance is not differentiable, we use the $L ^ { 1 }$ distance as an approximation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 OPTIMIZATION IN INPUT SPACE ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While in our method the optimization is done in the input space, in the previously mentioned AnoGAN, the search for the optimal reconstruction is done by iterating over $\\mathbf { z }$ samples with the ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "energy defined in equation 2. Following the aforementioned analogy between a GAN generator $G$ and a VAE decoder $D e c$ , a similar approach in the context of a VAE would be to use the energy ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e01f7b6f97ff6b949efb4bee6be337b31b796c1b87c053c831539f5827cbfe73.jpg",
+        "text": "$$\n| | \\mathbf { x } - D e c ( \\mathbf { z } ) | | _ { 1 } - \\lambda \\cdot \\log p ( \\mathbf { z } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the $- \\log p ( \\mathbf { z } )$ term has the same role as AnoGAN’s $| | f _ { D } ( \\mathbf { x } ) - f _ { D } ( G ( \\mathbf { z } ) ) | | _ { 1 }$ term, to ensure that $D e c ( \\mathbf { z } )$ stays within the learned manifold. We chose not to iterate over $\\mathbf { z }$ in the latent space for two reasons. First, because as noted in Dai & Wipf (2019) and Hoffman & Johnson (2016), the prior $p ( \\mathbf { z } )$ is not always a good proxy for the real image of the distribution in the latent space $q ( \\mathbf { z } )$ . Second, because the VAE tends to ignore some details of the original image in its reconstruction, considering that these details are part of the independent pixel noise allowed by the modeling of $p ( \\mathbf { x } | \\mathbf { z } )$ as a diagonal Gaussian, which causes its infamous blurriness. An optimization in latent space would have to recreate the high frequency structure of the image, whereas iterating over the input image space, and starting the descent on the input image $\\mathbf { x } _ { \\mathrm { 0 } }$ , allows us to keep that structure and thus to obtain projections of higher quality. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 OPTIMIZING GRADIENT DESCENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            315,
+            446,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We observed that using the Adam optimizer (Kingma & Ba, 2015) is beneficial for the quality of the optimization. Moreover, to speed up the convergence and further preserve the aforementioned high frequency structure of the input, we propose to compute our iterative samples using the pixel-wise reconstruction error of the VAE. To explain the intuition behind this improvement, we will consider the inpainting task. In this setting, as in anomaly localization, a local perturbation is added on top of a normal image. However, in the classic inpainting task, the localization of the perturbation is known beforehand, and we can use the localization mask $\\pmb { \\Omega }$ to only change the value of the anomalous pixels in the gradient descent: ",
+        "bbox": [
+            173,
+            340,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f23f9b783749f4859ba4a788ef18043a7df6f407c9b7dc834ea915047c2fa106.jpg",
+        "text": "$$\n\\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - \\alpha \\cdot \\left( \\nabla _ { \\mathbf { x } } E ( \\mathbf { x } _ { t } ) \\odot \\Omega \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            450,
+            617,
+            468
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ is the Hadamard product. ",
+        "bbox": [
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+            469,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For anomaly localization and blind inpainting, where this information is not available, we compute the pixel-wise reconstruction error which gives a rough estimate of the mask. The term $\\nabla _ { \\mathbf x } E ( \\mathbf x _ { t } )$ is therefore replaced with $\\nabla _ { \\mathbf x } E ( \\mathbf x _ { t } ) \\odot ( { \\mathbf x _ { t } } - f _ { V A E } ( \\mathbf { x } _ { t } ) ) ^ { 2 } .$ ) in equation 5: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af7ae15f47a600f88726caf7fdc9cf674f2a0aee385c0a8de68c0eafaf942287.jpg",
+        "text": "$$\n\\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - \\alpha \\cdot \\left( \\nabla _ { \\mathbf { x } } E ( \\mathbf { x } _ { t } ) \\odot ( \\mathbf { x } _ { t } - f _ { V A E } ( \\mathbf { x } _ { t } ) ) ^ { 2 } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            540,
+            671,
+            558
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { V A E } ( \\mathbf { x } )$ is the standard reconstruction of the VAE. Optimizing the energy this way, a pixel where the reconstruction error is high will update faster, whereas a pixel with good reconstruction will not change easily. This prevents the image to update its pixels where the reconstruction is already good, even with a high learning rate. As can be seen in appendix B, this method converges to the same performance as the method of equation 5, but with fewer iterations. An illustration of our method can be found in figure 2. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5 STOP CRITERION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A standard stop criterion based on the convergence of the energy can efficiently be used. Using the adversarial setting introduced in section 3.1, we also propose to stop the gradient descent when a certain predefined threshold on the VAE loss is reached. For example, such a threshold can be chosen to be a quantile of the empirical loss distribution computed on the training set. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate the proposed method for two different applications: anomaly segmentation and image inpainting. Both applications are interesting use cases of our method, where we search to reconstruct partially corrupted images, correcting the anomalies while preserving the uncorrupted image regions. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 UNSUPERVISED ANOMALY SEGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In order to evaluate the proposed method for the task of anomaly segmentation, we perform experiments with the recently proposed MVTec dataset (Bergmann et al., 2019). This collection of datasets ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Results for anomaly segmentation on MVTec datasets, expressed in AUROC (Area Under the Receiver Operating Characteristics). Four different baselines are trained on normal samples and are augmented by our proposed gradient based reconstruction (grad) for comparison: A deterministic autoencoder trained with $L ^ { \\tilde { 2 } }$ loss $( L ^ { 2 } \\mathbf { A E } )$ as in (Bergmann et al., 2019); A deterministic autoencoder trained with DSSIM loss (DSAE) as in (Bergmann et al., 2019); A variational autoencoder (VAE); And a variational autoencoder with a learned decoder variance $\\langle \\gamma \\cdot$ -VAE) as in (Dai & Wipf, 2019). For each result a green or red background denotes respectively an improvement or a decrease in performance compared to the baseline. It can be seen that the proposed gradient-based reconstruction achieves the best segmentation for most datasets, with a mean improvement rate of $9 . 5 2 \\%$ over all baselines. ",
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+        "page_idx": 5
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Category</td><td>L²AE</td><td>L²AE grad</td><td>DSAE</td><td>DSAE grad</td><td>VAE</td><td>VAE grad</td><td>γ-VAE</td><td>γ-VAE grad</td></tr><tr><td rowspan=\"5\">TJ annes</td><td>carpet</td><td>0.539</td><td>0.734</td><td>0.545</td><td>0.774</td><td>0.580</td><td>0.735</td><td>0.648</td><td>0.727</td></tr><tr><td>grid</td><td>0.960</td><td>0.981</td><td>0.960</td><td>0.980</td><td>0.888</td><td>0.961</td><td>0.950</td><td>0.979</td></tr><tr><td>leather</td><td>0.751</td><td>0.921</td><td>0.710</td><td>0.602</td><td>0.834</td><td>0.925</td><td>0.818</td><td>0.897</td></tr><tr><td>tile</td><td>0.476</td><td>0.575</td><td>0.496</td><td>0.626</td><td>0.465</td><td>0.654</td><td>0.491</td><td>0.581</td></tr><tr><td>wood</td><td>0.630</td><td>0.805</td><td>0.641</td><td>0.738</td><td>0.695</td><td>0.838</td><td>0.665</td><td>0.809</td></tr><tr><td rowspan=\"9\">seetqt</td><td>bottle</td><td>0.909</td><td>0.916</td><td>0.933</td><td>0.951</td><td>0.902</td><td>0.922</td><td>0.913</td><td>0.931</td></tr><tr><td>cable</td><td>0.732</td><td>0.864</td><td>0.790</td><td>0.859</td><td>0.828</td><td>0.910</td><td>0.777</td><td>0.880</td></tr><tr><td>capsule</td><td>0.786</td><td>0.952</td><td>0.769</td><td>0.884</td><td>0.862</td><td>0.917</td><td>0.814</td><td>0.917</td></tr><tr><td>hazelnut</td><td>0.976</td><td>0.984</td><td>0.966</td><td>0.966</td><td>0.977</td><td>0.976</td><td>0.977</td><td>0.988</td></tr><tr><td>metalnut</td><td>0.880</td><td>0.899</td><td>0.881</td><td>0.920</td><td>0.881</td><td>0.907</td><td>0.883</td><td>0.914</td></tr><tr><td>pill</td><td>0.885</td><td>0.912</td><td>0.895</td><td>0.927</td><td>0.888</td><td>0.930</td><td>0.897</td><td>0.935</td></tr><tr><td>screw</td><td>0.979</td><td>0.980</td><td>0.983</td><td>0.925</td><td>0.958</td><td>0.945</td><td>0.976</td><td>0.972</td></tr><tr><td>toothbrush</td><td>0.971</td><td>0.983</td><td>0.973</td><td>0.984</td><td>0.971</td><td>0.985</td><td>0.971</td><td>0.983</td></tr><tr><td>transistor</td><td>0.906</td><td>0.921</td><td>0.904</td><td>0.934</td><td>0.894</td><td>0.919</td><td>0.896</td><td>0.931</td></tr><tr><td>zipper</td><td></td><td>0.680</td><td>0.889</td><td>0.828</td><td>0.887</td><td>0.814</td><td>0.869</td><td>0.706</td><td>0.871</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "consists of 15 different categories of objects and textures in the context of industrial inspection, each category containing a number of normal and anomalous samples. ",
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+        "type": "text",
+        "text": "We train our model on normal training samples and test it on both normal and anomalous test samples to evaluate the anomaly segmentation performance. ",
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+        "type": "text",
+        "text": "We perform experiments with three different baseline autoencoders: A “vanilla” variational autoencoder with decoder covariance matrix fixed to identity (Kingma & Welling, 2014), a variational autoencoder with learned decoder variance (Dai & Wipf, 2019), a “vanilla” deterministic autoencoder trained with $L ^ { 2 }$ as reconstruction loss $( L ^ { 2 } \\mathrm { A E } )$ and a deterministic autoencoder trained with DSSIM reconstruction loss (DSAE), as proposed by Bergmann et al. (2018). For the sake of a fair comparison, all the autoencoder models are parameterized by convolutional neural networks with the same architecture, latent space dimensionality (set to 100), learning rate (set to 0.0001) and number of epochs (set to 300). The architecture details (layers, paddings, strides) are the same as described in Bergmann et al. (2018) and Bergmann et al. (2019). Similarly to the authors in Bergmann et al. (2019), for the textures datasets, we first subsample the original dataset images to $5 1 2 \\times 5 1 2$ and then crop random patches of size $1 2 8 \\times 1 2 8$ which are used to train and test the different models. For the object datasets, we directly subsample the original dataset images to $1 2 8 \\times 1 2 8$ unlike in Bergmann et al. (2019) who work on $2 5 6 \\times 2 5 6$ images, then we perform rotation and translation data augmentations. For all datasets we train on 10000 images. ",
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+    {
+        "type": "text",
+        "text": "Anomaly segmentation is then computed by reconstructing the anomalous image and comparing it with the original. We perform the comparison between reconstructed and original with the DSSIM metric as it has been observed in Bergmann et al. (2018) that it provides better anomaly localization than $L ^ { 2 }$ or $L ^ { 1 }$ distances. For the gradient descent, we set the step size $\\alpha : = 0 . 5$ , $L ^ { 1 }$ regularization weight $\\lambda : = 0 . 0 5$ and the stop criterion is achieved when a sample reconstruction loss is inferior to the minimum reconstruction loss over the training set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In table 1 we show the AUROC (Area Under the Receiver Operating Characteristics) for different autoencoder methods, with different thresholds applied to the DSSIM anomaly map computed between original and reconstructed images. Note that an AUROC of 1 expresses the best possible segmentation in terms of normal and anomalous pixels. For each autoencoder variant we compare the baseline reconstruction with the proposed gradient-based reconstruction (grad.). As in Bergmann et al. (2019) we observe that an overall best model is hard to identify, however we show that our method increases the AUC values for almost all autoencoder variants. Aggregating the results over all datasets and baselines, we report a mean improvement rate of $9 . 5 2 \\%$ , with a median of $4 . 3 3 \\%$ , a 25th percentile of $1 . 8 6 \\%$ , and a 75th percentile of $1 5 . 8 6 \\%$ . The histogram of the improvement rate for all datasets and baselines is provided in appendix F, as well as a short analysis. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In figure 3 we compare our anomaly segmentation with a baseline $L ^ { 2 }$ autoencoder Bergmann et al. (2019) $( L ^ { 2 } \\mathbf { A E } )$ for a number of image categories. For all results in figure 3, we set the same threshold to 0.2 to the anomaly detection map given by the DSSIM metric. The visual results in figure 3 highlights an overall improvement of anomaly localization by our proposed iterative reconstruction $\\bar { L ^ { 2 } } \\mathrm { A } \\bar { \\mathrm { E } }$ -grad). See appendix C for additional visual results of anomaly segmentation on remaining categories of MVTec dataset, and on remaining baseline models. ",
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+    {
+        "type": "image",
+        "img_path": "images/65dae72096b01be8ba39002ab1ce20d45dd35bf9bf93522d3aa43aca83d05f6c.jpg",
+        "image_caption": [
+            "Figure 3: First row: Normal samples of hazelnut, grid, cable, wood, carpet and bottle categories in MVTec dataset; Second row: anomalous samples from the aforementioned dataset categories; Third row: Anomaly segmentation with baseline $\\bar { L } ^ { 2 }$ autoencoder (Bergmann et al., 2019); Fourth row: our proposed anomaly segmentation with $L ^ { 2 }$ autoencoder augmented with gradient-based iterative reconstruction. Ground truth is represented by red contour, and each estimated segmentation by a green overlay. It can be seen that anomaly segmentation is refined by our proposed method, with a tendency of detecting less false positives. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2 INPAINTING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Image inpainting is a well known image reconstruction problem which consists of reconstructing a corrupted or missing part of an image, where the region to be reconstructed is usually given by a known mask. Many different approaches for inpainting have been proposed in the literature, such as anisotropic diffusion (Bertalmio et al., 2000), patch matching (Criminisi et al., 2004), context autoencoders (Pathak et al., 2016) and conditional variational autoencoders (Ivanov et al., 2019). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "If we consider that the region to be reconstructed is not known beforehand, the problem is sometimes called blind inpainting (Altinel et al., 2018), and the corrupted part can be seen as an anomaly to be corrected. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/3c525dc352b059d364d4bd4eb4992154d606ff497f94c348f4d97f8db18a4df2.jpg",
+        "image_caption": [
+            "Figure 4: Inpainting experiment performed on CelebA dataset, where the test face images are masked with uniform noise. The baseline VAE reconstruction is disturbed by the noise mask, providing a poor inpainting. The proposed gradient-based VAE provides a more convincing inpainting by an iterative process. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We performed experiments with image inpainting on the CelebA dataset (Liu et al., 2015), which consists of celebrity faces. In figure 4 we compare the inpainting results obtained with a baseline VAE with learned variance $\\left( \\gamma { \\mathrm { - V A E } } \\right)$ and Resnet architecture, as described by Dai & Wipf (2019), with the same VAE model, augmented by our proposed gradient-based iterative reconstruction. Note that for the regular inpainting task, gradients are multiplied by the inpainting mask at each iteration (equation 7), while for the blind inpainting task, the mask is unknown. See appendix D for a comparison with a recent method based on variational autoencoders, proposed by Ivanov et al. (2019). ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Baur et al. (2018) have used autoencoder reconstructions to localize anomalies in MRI scans, and have compared several variants using diverse per-pixel distances as well as perceptual metrics derived from a GAN-like architecture. Bergmann et al. (2018) use the structural similarity metric (Wang et al., 2004) to compare the original image and its reconstruction to achieve better anomaly localization, and also presents the SSIM autoencoder, which is trained directly with this metric. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Zimmerer et al. (2018) use the derivative of the VAE loss function with respect to the input, called the score. The amplitude of the score is supposed to indicate how abnormal a pixel is. While we agree that the gradient of the loss is an indication of an anomaly, we think that we have to integrate this gradient over the path from the input to the normal manifold to obtain meaningful information. We compare our results to score-based results for anomaly localization in appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The work that is the most related to ours is AnoGAN (Schlegl et al., 2017). We have mentioned above the differences between the two approaches, which, apart from the change in underlying architectures, boil down to the ability in our method to update directly the input image instead of searching for the optimal latent code. This enables the method to converge faster and above all to keep higher-frequency structures of the input, which would have been deteriorated if it were passed through the AE bottleneck. Bergmann et al. (2019) compare standard AE reconstructions techniques to AnoGAN, and observes that AnoGAN’s performances on anomaly localizations tasks are poorer than AE’s due to the mode collapse tendency of GAN architectures. Interestingly, updates on AnoGAN such as fast AnoGAN (Schlegl et al., 2019) or AnoVAEGAN (Baur et al., 2018) replaced the gradient descent search of the optimal z with a learned encoder model, yielding an approach very similar to the standard VAE reconstruction-based approaches, but with a reconstruction loss learned by a discriminator, which is still prone to mode collapse (Thanh-Tung et al., 2019). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed a novel method for unsupervised anomaly localization, using gradient descent of an energy defined by an autoencoder reconstruction loss. Starting from a sample under test, we iteratively update this sample to reduce its autoencoder reconstruction error. This method offers a way to incorporate human priors into what is the optimal projection of an out-of-distribution sample into the normal data manifold. In particular, we use the pixel-wise reconstruction error to modulate the gradient descent, which gives impressive anomaly localization results in only a few iterations. Using gradient descent in the input data space, starting from the input sample, enables us to overcome the autoencoder tendency to provide blurry reconstructions and keep normal high frequency structures. This significantly reduces the number of pixels that could be wrongly classified as defects when the autoencoder fails to reconstruct high frequencies. We showed that this method, which can easily be added to any previously trained autoencoder architecture, gives state-of-the-art results on a variety of unsupervised anomaly localization datasets, as well as qualitative reconstructions on an inpainting task. Future work can focus on replacing the $L ^ { 1 }$ -based regularization term with a Bayesian prior modeling common types of anomalies, and on further improving the speed of the gradient descent. ",
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+        "page_idx": 8
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+    {
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+    {
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+        "text": "Hoang Thanh-Tung, Truyen Tran, and Svetha Venkatesh. Improving generalization and stability of generative adversarial networks. CoRR, abs/1902.03984, 2019. ",
+        "bbox": [
+            168,
+            604,
+            823,
+            635
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhou Wang, Alan Bovik, Hamid Sheikh, and Eero Simoncelli. Image quality assessment: From error visibility to structural similarity. Image Processing, IEEE Transactions on, 13:600 – 612, 05 2004. doi: 10.1109/TIP.2003.819861. ",
+        "bbox": [
+            173,
+            642,
+            825,
+            685
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "David Zimmerer, Fabian Isensee, Jens Petersen, Simon Kohl A. A., and Klaus H. Maier-Hein. Unsupervised anomaly localization using variational auto-encoders. CoRR, abs/1907.02796, 2019. ",
+        "bbox": [
+            173,
+            746,
+            823,
+            775
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A COMPARISON WITH ZIMMERER ET AL. (2019) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            593,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/ce8b71adc930320ccbede62df90df18c9c46e4b54bcdec76e5f67e12f74c8c3e.jpg",
+        "table_caption": [
+            "Table 2: Complementary results for anomaly segmentation on MVTec datasets, expressed in AUROC for different pixel-wise scores derived from a baseline VAE: from left to right, squared error reconstruction $| | { \\bf x } - f _ { V A E } ( { \\bf x } ) | | ^ { 2 }$ (denoted $\\mathcal { L } _ { r } ( \\mathbf { x } ) )$ , gradient of the loss $| \\nabla _ { \\mathbf x } \\mathcal L ( \\mathbf x ) |$ , combination of both, gradient of the KL divergence $| D _ { \\mathrm { K L } } ( q ( \\mathbf { z } | \\mathbf { x } ) \\| p ( \\mathbf { z } ) )$ (denoted $| \\nabla _ { \\mathbf { x } } \\mathcal { L } _ { K L } ( \\mathbf { x } ) | )$ as well as combination of KL derivative and error reconstruction as suggested in Zimmerer et al. (2018; 2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Category</td><td>Lr(x)</td><td>VxL(x)</td><td>IVxL(x)) OLr(x)</td><td>IVxLKL(x)l</td><td>IVxLKL(x)) Lr(x)</td><td>VAE-grad</td></tr><tr><td rowspan=\"5\">TJssnets</td><td>carpet</td><td>0.537</td><td>0.580</td><td>0.566</td><td>0.553</td><td>0.555</td><td>0.735</td></tr><tr><td>grid</td><td>0.823</td><td>0.635</td><td>0.812</td><td>0.507</td><td>0.790</td><td>0.961</td></tr><tr><td>leather</td><td>0.783</td><td>0.650</td><td>0.792</td><td>0.627</td><td>0.791</td><td>0.925</td></tr><tr><td>tile</td><td>0.547</td><td>0.606</td><td>0.581</td><td>0.623</td><td>0.588</td><td>0.654</td></tr><tr><td>wood</td><td>0.686</td><td>0.691</td><td>0.726</td><td>0.643</td><td>0.713</td><td>0.838</td></tr><tr><td rowspan=\"10\">seetqt</td><td>bottle</td><td>0.831</td><td>0.762</td><td>0.832</td><td>0.629</td><td>0.830</td><td>0.922</td></tr><tr><td>cable</td><td>0.831</td><td>0.796</td><td>0.846</td><td>0.674</td><td>0.841</td><td>0.910</td></tr><tr><td>capsule</td><td>0.765</td><td>0.754</td><td>0.772</td><td>0.642</td><td>0.795</td><td>0.917.</td></tr><tr><td>hazelnut</td><td>0.907</td><td>0.831</td><td>0.908</td><td>0.468</td><td>0.885</td><td>0.976</td></tr><tr><td>metalnut</td><td>0.833</td><td>0.831</td><td>0.870</td><td>0.710</td><td>0.834</td><td>0.907</td></tr><tr><td>pill</td><td>0.869</td><td>0.833</td><td>0.872</td><td>0.480</td><td>0.826</td><td>0.930</td></tr><tr><td>screw</td><td>0.851</td><td>0.726</td><td>0.842</td><td>0.412</td><td>0.795</td><td>0.945</td></tr><tr><td>toothbrush</td><td>0.942</td><td>0.798</td><td>0.943</td><td>0.619</td><td>0.939</td><td>0.985</td></tr><tr><td>transistor</td><td>0.788</td><td>0.843</td><td>0.834</td><td>0.801</td><td>0.836</td><td>0.919</td></tr><tr><td>zipper</td><td>0.725</td><td>0.674</td><td>0.729</td><td>0.562</td><td>0.727</td><td>0.869</td></tr></table>",
+        "bbox": [
+            173,
+            227,
+            828,
+            468
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zimmerer et al. (2019) proposed to perform anomaly localization using different scores derived from the gradient of the VAE loss. In particular, it has been shown that the product of the VAE reconstruction error with the gradient of the KL divergence was very informative for medical images. In table 2 we compare the pixel-wise anomaly detection AUROC of these different scores with our method. For all experiments, we use the same “vanilla” VAE as described in section 4.1. ",
+        "bbox": [
+            173,
+            488,
+            825,
+            559
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "It can be seen that other VAE-based methods using a single evaluation of the gradient are constantly outperformed by our method. ",
+        "bbox": [
+            174,
+            565,
+            823,
+            593
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B CONVERGENCE SPEED ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            397,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/345cea3215d5de240941e391b07b8e8c6ecd6b2a51cb52dde3287447ce2a70b3.jpg",
+        "image_caption": [
+            "Figure 5: Evolution of pixel-wise anomaly detection AUROC performance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            142,
+            751,
+            377
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In figure 5 we compare the number of iterations needed to reach convergence with our two proposals for gradient descent: Standard update as in equation 5 and Tuned update using a gradient mask computed with the VAE reconstruction error, as in equation 8. The model is a VAE with learned decoder variance (Dai & Wipf, 2019), trained on the Grid dataset (Bergmann et al., 2019). We compute the mean pixel-wise anomaly detection AUROC after each iteration on the test set. ",
+        "bbox": [
+            173,
+            426,
+            825,
+            496
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We can see that the tuned method converges to the same performance as the standard method, with far fewer iterations. ",
+        "bbox": [
+            173,
+            502,
+            823,
+            531
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/2cebada9b7197e00c2fab69f1850ebdeacfeefa2a889bd15909f9a0d95c80f61.jpg",
+        "image_caption": [
+            "Figure 6: From left to right: Normal; Anomalous; Anomaly segmentation with baseline $L ^ { 2 }$ autoencoder (Bergmann et al., 2019); Our proposed anomaly segmentation with $L ^ { 2 }$ autoencoder augmented with gradient-based iterative reconstruction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            135,
+            666,
+            713
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a930ed3a0d4bf482f1bb30b17161e0bf7c1b209bebf56b1660f760c38e4db58.jpg",
+        "image_caption": [
+            "Figure 7: Illustration of anomaly localization comparison over four baselines ( $L ^ { 2 } { \\mathrm { A E } }$ , DSAE, VAE, $\\gamma$ -VAE). Ground truth is represented by red contour, and each estimated segmentation by a green overlay. It can be seen that anomaly segmentation is overall improved when different baselines are augmented by our proposed gradient descent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            103,
+            779,
+            696
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D INPAINTING COMPARISON ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            426,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/ffc49efc49762ecb3f4c500413e7267bb1d6f339702e0013a7cde29f0684c8c5.jpg",
+        "image_caption": [
+            "Figure 8: Inpainting comparison. Each batch is made of four rows, from top to bottom: Masked input image; VAE with arbitrary conditioning (VAEAC, Ivanov et al. (2019)); Ours; Ground truth. The quality of the reconstructions is comparable, even though our VAE is trained without any assumptions over the mask’s properties. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            131,
+            777,
+            720
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/2a43dfab4dcfee18aabf106a89fa776779a096451340a5174bf902c8a544094a.jpg",
+        "image_caption": [
+            "Figure 9: Principle of the energy optimization to project anomalous sample on the normal manifold "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            135,
+            727,
+            404
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 9 illustrates our method principle. We start with a defective input $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ whose reconstruction $\\scriptstyle { \\hat { \\mathbf { x } } } _ { 0 }$ does not necessarily lie on the normal data manifold. As the optimization process carries on, the optimized sample $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ and its reconstruction look more similar and get closer to the manifold. The regularization term of the energy function makes sure that the optimized sample stays close to the original sample. ",
+        "bbox": [
+            173,
+            452,
+            825,
+            522
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/af15f62eeb6a6ad21ae17c81d2036303dc40784f6c062bc7493c7a6a7a566a04.jpg",
+        "image_caption": [
+            "Figure 10: Distribution of the improvement rate over all presented baselines and all datasets in MVTec AD. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            171,
+            692,
+            448
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 shows the distribution of the AUC improvement rate over all presented baselines and all datasets in MVTec AD using our gradient-based projection method. ",
+        "bbox": [
+            173,
+            518,
+            823,
+            547
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4cf2b573cd1eb684730e0d7d66fb3e96c26a314fb4f70909667f64e890f8a49e.jpg",
+        "text": "$$\n{ \\mathrm { i m p r o v e m e n t r a t e } } = { \\frac { A U C _ { g r a d } - A U C _ { b a s e } } { A U C _ { b a s e } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            553,
+            645,
+            587
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "• $8 . 3 \\%$ of data points are under the 0 value delimiting an increase or decrease in AUC due to our method, and $9 1 . 7 \\%$ data points are over this value. Our method increases the AUC in a vast majority of cases. ",
+        "bbox": [
+            214,
+            595,
+            825,
+            637
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "• The median is at $4 . 3 3 \\%$ , the 25th percentile at $1 . 8 6 \\%$ , and the 75th percentile at $1 5 . 8 6 \\%$ . ",
+        "bbox": [
+            215,
+            642,
+            813,
+            659
+        ],
+        "page_idx": 16
+    }
+]

parse/train/M_lkFOwVdYc/M_lkFOwVdYc.md

@@ -0,0 +1,260 @@
+# Long-Short Transformer: Efficient Transformers for Language and Vision
+
+Chen $Z \mathrm { { h u } ^ { \ddag 1 * } }$ , Wei Ping†2, Chaowei Xiao†2,3, Mohammad Shoeybi2, Tom Goldstein1, Anima Anandkumar2,4, and Bryan Catanzaro2
+
+1University of Maryland, College Park 2 NVIDIA 3Arizona State University 4California Institute of Technology ‡chenzhu@cs.umd.edu, †{wping, chaoweix}@nvidia.com
+
+# Abstract
+
+Transformers have achieved success in both language and vision domains. However, it is prohibitively expensive to scale them to long sequences such as long documents or high-resolution images, because self-attention mechanism has quadratic time and memory complexities with respect to the input sequence length. In this paper, we propose Long-Short Transformer (Transformer-LS), an efficient self-attention mechanism for modeling long sequences with linear complexity for both language and vision tasks. It aggregates a novel long-range attention with dynamic projection to model distant correlations and a short-term attention to capture fine-grained local correlations. We propose a dual normalization strategy to account for the scale mismatch between the two attention mechanisms. Transformer-LS can be applied to both autoregressive and bidirectional models without additional complexity. Our method outperforms the state-of-the-art models on multiple tasks in language and vision domains, including the Long Range Arena benchmark, autoregressive language modeling, and ImageNet classification. For instance, Transformer-LS achieves 0.97 test BPC on enwik8 using half the number of parameters than previous method, while being faster and is able to handle $3 \times$ as long sequences compared to its full-attention version on the same hardware. On ImageNet, it can obtain the state-of-the-art results (e.g., a moderate size of $5 5 . 8 \mathbf { M }$ model solely trained on $2 2 4 \times 2 2 4$ ImageNet-1K can obtain Top-1 accuracy $8 4 . 1 \%$ ), while being more scalable on high-resolution images. The source code and models are released at https://github.com/NVIDIA/transformer-ls.
+
+# 1 Introduction
+
+Transformer-based models [1] have achieved great success in the domains of natural language processing (NLP) [2, 3] and computer vision [4–6]. These models benefit from the self-attention module, which can capture both adjacent and long-range correlations between tokens while efficiently scaling on modern hardware. However, the time and memory consumed by self-attention scale quadratically with the input length, making it very expensive to process long sequences. Many language and vision tasks benefit from modeling long sequences. In NLP, document-level tasks require processing long articles [e.g., 7, 8], and the performance of language models often increases with sequence length [e.g., 9, 10]. In computer vision, many tasks involve high-resolution images, which are converted to long sequences of image patches before being processed with Transformer models [4, 6, 11]. As a result, it is crucial to design an efficient attention mechanism for long sequence modeling that generalizes well across different domains.
+
+Various methods have been proposed to reduce the quadratic cost of full attention. However, an efficient attention mechanism that generalizes well in both language and vision domains is less explored. One family of methods is to sparsify the attention matrix with predefined patterns such as sliding windows [e.g., 12–15] and random sparse patterns [16]. These methods leverage strong inductive biases to improve both computational and model performance, but they limit the capacity of a self-attention layer because each specific token can only attend to a subset of tokens. Another family of methods leverages low-rank projections to form a low resolution representation of the input sequence, but the successful application of these methods has been limited to certain NLP tasks [e.g., 17–19]. Unlike sparse attention, this family of methods allows each token to attend to the entire input sequence. However, due to the loss of high-fidelity token-wise information, their performance sometimes is not as good as full attention or sparse attention on tasks that require fine-grained local information, including standard benchmarks in language [20] and vision [21].
+
+Despite the rapid progress in efficient Transformers, some proposed architectures can only be applied to bidirectional models [e.g., 15, 16, 18]. Transformer-based autoregressive models have achieved great successes in language modeling [22], image synthesis [23], and text-to-image synthesis [24], which also involve long texts or high-resolution images. It is desirable to design an efficient transformer that can be applied to both autoregressive and bidirectional models.
+
+In this work, we unify a local window attention and a novel long-range attention into a single efficient attention mechanism. We show that these two kinds of attention have complementary effects that together yield the state-of-the-art results on a range of tasks in language and vision, for both autoregressive and bidirectional models. Specifically, we make the following contributions:
+
+• We propose Long-Short Transformer (Transformer-LS), an efficient Transformer that integrates a dynamic projection based attention to model long-range correlations, and a local window attention to capture fine-grained correlations. Transformer-LS can be applied to both autoregressive and bidirectional models with linear time and memory complexity.   
+• We compute a dynamic low-rank projection, which depends on the content of the input sequence. In contrast to previous low-rank projection methods, our dynamic projection method is more flexible and robust to semantic-preserving positional variations (e.g., insertion, paraphrasing). We demonstrate that it outperforms previous low-rank methods [17, 18] on Long Range Arena benchmark [20].   
+• We identify a scale mismatch problem between the embeddings from the long-range and shortterm attentions, and design a simple but effective dual normalization strategy, termed DualLN, to account for the mismatch and enhance the effectiveness of the aggregation.   
+• We demonstrate that Long-Short Transformer, despite its low memory and runtime complexity, outperforms the state-of-the-art models on a set of tasks from Long Range Arena, and autoregressive language modeling on enwik8 and text8. In addition, the proposed efficient attention mechanism can be easily applied to the most recent vision transformer architectures [6, 11] and provides state-of-the-art results, while being more scalable to high-resolution images. We also investigate the robustness properties of the Transformer-LS on diverse ImageNet datasets.
+
+# 2 Related Work
+
+# 2.1 Efficient Transformers
+
+In recent years, many methods have been introduced for dealing with the quadratic cost of full attention. In general, they can be categorized as follows: i) Sparse attention mechanism with predefined patterns (e.g., sliding window), including Sparse Transformer [12], Image Transformer [13], Axial Transformer [25] for modeling images, and Longformer [14], blockwise self-attention [26], ETC [15], Big Bird [16] for modeling language. ii) Low-rank projection attention, including Linformer [17], Nyströmformer [18], Synthesizer [19]. For example, Linformer uses linear layers to project the original high resolution keys $( K )$ and values $( V )$ with length $n$ to low resolution with size $r$ $( r \ll n )$ ) and allows all query tokens $( Q )$ to attend these compressed representations. iii) Memory-based mechanisms like Compressive Transformer [10] and Set Transformer [27], which use extra memories for caching global long-range information for use in computing attention between distant tokens. iv) Kernel-based approximation of the attention matrix, including Performer [28], Linear
+
+![](images/cac6035bb2eb7c5eb4e197f0f078ccb16d4a2c66c0c57d273bbe65a2ec761215.jpg)  
+Figure 1: Long-short term attention of a single attention head. Here, the sequence length $n = 8$ , hidden dimension $d = 3$ , local window segment size $w = 2$ , and rank of dynamic projection $r = 3$ . Within the figure, $K ( V )$ denotes key $K$ or value $V$ . In the left figure, we virtually replicate $K$ or $V \in \mathbb { R } ^ { n \times d }$ into $n$ rows, and highlight the keys and values within the attention span (denoted as $\tilde { K } ( \tilde { V } ) )$ of all $_ n$ queries $Q$ for the short-term attention. In the middle figure, all queries attend to the same projected keys $\bar { K }$ and values $\bar { V }$ within the long-term attention. In the right figure, $\tilde { K } ( \tilde { V } )$ and $\bar { K } ( \bar { V } )$ are first normalized with two sets of LayerNorms, and the queries attend to normalized $\tilde { K } ( \tilde { V } )$ and $\bar { K } ( \bar { V } )$ within their attention span simultaneously.
+
+Transformer [29], and Random Feature Attention [30]. vi) Similarity and clustering based methods, including Reformer [31], Routing Transformer [32], and Sinkhorn Transformer [33].
+
+Our method seamlessly integrates both low-rank projection and local window attentions, to leverage their strengths for modeling long-range and short-term correlations. In particular, our long-range attention uses a dynamic low-rank projection to encode the input sequence, and outperforms the previous low-rank projection method used by the Linformer [17]. In the similar vein, a few other methods also try to combine the strengths of different methods. For example, Longformer [14] and ETC [15] augment local window attention with task motivated global tokens. Such global tokens may not be applicable for some tasks (e.g., autoregressive modelling). BigBird [16] further combines local window and global token attention with random sparse attention. It is not applicable in autoregressive tasks because the global token and random sparse pattern are introduced. To compress the model footprint on edge devices, Lite Transformer [34] combines convolution and self-attention, but it still has quadratic complexity for long sequences.
+
+# 2.2 Vision Transformers
+
+Vision Transformer (ViT) [4] splits images as small patches and treats the patches as the input word tokens. It uses a standard transformer for image classification and has shown to outperform convolutional neural networks (e.g., ResNet [35]) with sufficient training data. DeiT [36] has applied the teacher-student strategy to alleviate the data efficiency problem of ViT and has shown strong comparable performance using only the standard ImageNet dataset [37]. Instead of applying transformer at a single low resolution of patches (e.g., $1 6 \times 1 6$ patches), very recent works, including Pyramid Vision Transformer (PVT) [5], Swin-Transformer [38], T2T-ViT [39], Vision Longformer (ViL) [11] and Convolutional Vision Transformer (CvT) [6], stack a pyramid of ViTs to form a multi-scale architecture and model long sequences of image patches at much higher resolution (e.g., $5 6 \times 5 6 = 3 1 3 6$ patches for images with $2 2 4 \times 2 2 4$ pixels). Most of these methods have quadratic complexity of self-attention with respect to the input image size.
+
+To reduce the complexity, Swin-Transformer [38] achieves linear complexity by limiting the computation of self-attention only within each local window. HaloNet [40] applies local attention on blocked images and only has quadratic complexity with respect to the size of the block. Perceiver [41] uses cross-attention between data and latent arrays to replace the self-attention on data to remove the quadratic complexity bottleneck. Vision Longformer (ViL) [11], another concurrent work, achieves linear complexity by adapting Longformer [14] to Vision. ViL augments local window attention with task-specific global tokens, but the global tokens are not applicable for decoding task (e.g., image synthesis [23, 24]). In contrast, our method reduces the quadratic cost to linear cost by combining local window attention with global dynamic projection attention, which can be applied to both encoding and decoding tasks.
+
+# 3 Long-Short Transformer
+
+Transformer-LS approximates the full attention by aggregating long-range and short-term attentions, while maintaining its ability to capture correlations between all input tokens. In this section, we first introduce the preliminaries of multi-head attention in Transformer. Then, we present the short-term attention via sliding window, and long-range attention via dynamic projection, respectively. After that, we propose the aggregating method and dual normalization (DualLN) strategy. See Figure 1 for an illustration of our long-short term attention.
+
+# 3.1 Preliminaries and Notations
+
+Multi-head attention is a core component of the Transformer [1], which computes contextual representations for each token by attending to the whole input sequence at different representation subspaces. It is defined as
+
+$$
+\mathrm { M u l t i H e a d } ( Q , K , V ) = \mathrm { C o n c a t } ( H _ { 1 } , H _ { 2 } , . . . , H _ { h } ) W ^ { O } ,
+$$
+
+where $Q , K , V \in \mathbb { R } ^ { n \times d }$ are the query, key and value embeddings, $W ^ { O } \in \mathbb { R } ^ { d \times d }$ is the projection matrix for output, the $i$ -th head $\bar { H _ { i } } \in \bar { \mathbb { R } } ^ { n \times \check { d } _ { k } }$ is the scaled dot-product attention, and $d _ { k } = d / h$ is the embedding dimension of each head,
+
+$$
+H _ { i } = \mathrm { A t t e n t i o n } ( Q W _ { i } ^ { Q } , K W _ { i } ^ { K } , V W _ { i } ^ { V } ) = \mathrm { s o f t m a x } \left[ \frac { Q W _ { i } ^ { Q } \left( K W _ { i } ^ { K } \right) ^ { \mathsf { T } } } { \sqrt { d _ { k } } } \right] V W _ { i } ^ { V } = A _ { i } V W _ { i } ^ { V } ,
+$$
+
+where $W _ { i } ^ { Q } , W _ { i } ^ { K } , W _ { i } ^ { V } \in \mathbb { R } ^ { d \times d _ { k } }$ are learned projection matrices, and $A _ { i } \in \mathbb { R } ^ { n \times n }$ denotes the full attention matrix for each attention head. The complexity of computing and storing $A _ { i }$ is $O ( n ^ { 2 } )$ , which can be prohibitive when $n$ is large. For simplicity, our discussion below is based on the case of 1D input sequences. It is straightforward to extend to the 2D image data given a predetermined order.
+
+# 3.2 Short-term Attention via Segment-wise Sliding Window
+
+We use the simple yet effective sliding window attention to capture fine-grained local correlations, where each query attends to nearby tokens within a fixed-size neighborhood. Similar techniques have also been adopted in [14, 16, 11]. Specifically, we divide the input sequence into disjoint segments with length $w$ for efficiency reason. All tokens within a segment attend to all tokens within its home segment, as well as $w / 2$ consecutive tokens on the left and right side of its home segment (zero-padding when necessary), resulting in an attention span over a total of $2 w$ key-value pairs. See Figure 5 in Appendix for an illustration. For each query $Q _ { t }$ at the position $t$ within the $i$ -th head, we denote the $2 w$ key-value pairs within its window as $\bar { K } _ { t } , \tilde { V } _ { t } \in \mathbb { R } ^ { \bar { 2 w } \times d }$ . For implementation with PyTorch, this segment-wise sliding window attention is faster than the per-token sliding window attention where each token attends to itself and $w$ tokens to its left and right, and its memory consumption scales linearly with sequence length; see [14] and our Figure 3 for more details.
+
+The sliding window attention can be augmented to capture long-range correlations in part, by introducing different dilations to different heads of sliding window attention [14]. However, the dilation configurations for different heads need further tuning and an efficient implementation of multi-head attention with different dilations is non-trivial. A more efficient alternative is to augment sliding window attention with random sparse attention [16], but this does not guarantee that the long-range correlations are captured in each layer as in full attention. In the following section, we propose our long-range attention to address this issue.
+
+# 3.3 Long-range Attention via Dynamic Projections
+
+Previous works have shown that the self-attention matrix can be well approximated by the product of low-rank matrices [17]. By replacing the full attention with the product of low-rank matrices [42, 19, 18, 43, 28], each query is able to attend to all tokens. Linformer [17] is one of the most representative models in this category. It learns a fixed projection matrix to reduce the length of the keys and values, but the fixed projection is inflexible to semantic-preserving positional variations.
+
+Starting from these observations, we parameterize the dynamic low-rank projection at $i$ -th head as $P _ { i } = f ( K ) \in \mathbb { R } ^ { n \times r }$ , where $r \ll n$ is the low rank size and $P _ { i }$ depends on all the keys $K \in \mathbb { R } ^ { n \times d }$ of input sequence. It projects the $( n \times d _ { k } )$ -dimensional key embeddings $K W _ { i } ^ { K }$ and value embeddings $V W _ { i } ^ { V }$ into shorter, $( r \times d _ { k } )$ -dimensional key ${ \bar { K } } _ { i }$ and value $\bar { V } _ { i }$ embeddings. Unlike Linformer [17], the low-rank projection matrix is dynamic, which depends on the input sequence and is intended to be more flexible and robust to, e.g., insertion, deletion, paraphrasing, and other operations that change sequence length. See Table 2 for examples. Note that, the query embeddings $Q \bar { W } _ { i } ^ { Q } \in \mathbb { R } ^ { n \times d _ { k } }$ are kept at the same length, and we let each query attend to ${ \bar { K } } _ { i }$ and $\bar { V } _ { i }$ . In this way, the full $( n \times n )$ attention matrix can be decomposed into the product of two matrices with $r$ columns or rows. Specifically, we define the dynamic projection matrix $P _ { i } \in \mathbb { R } ^ { n \times r }$ and the key-value embeddings $\bar { K } _ { i } , \dot { \bar { V } } _ { i } \in \mathbb { R } ^ { r \times \mathbf { \dot { d } } _ { k } }$ of low-rank attention as
+
+$$
+P _ { i } = \mathrm { s o f t m a x } ( K W _ { i } ^ { P } ) , \bar { K } _ { i } = P _ { i } ^ { \intercal } K W _ { i } ^ { K } , \bar { V } _ { i } = P _ { i } ^ { \intercal } V W _ { i } ^ { V } ,
+$$
+
+where $W _ { i } ^ { P } \in \mathbb { R } ^ { d \times r }$ are learnable parameters,2 and the softmax normalizes the projection weights on the first dimension over all $n$ tokens, which stabilizes training in our experiments. Note that $K = V$ in all the experiments we have considered, so $P _ { i }$ remains the same if it depends on $V$ . The computational complexity of Eq. 3 is $O ( r n )$ .
+
+To see how the full attention is replaced by the product of low-rank matrices, we compute each head $H _ { i } \in \mathbb { R } ^ { n \times d _ { k } }$ of long-range attention as,
+
+$$
+\bar { H } _ { i } = \underbrace { \mathrm { s o f t m a x } \left[ \frac { Q W _ { i } ^ { Q } \bar { K } _ { i } ^ { \top } } { \sqrt { d _ { k } } } \right] } _ { \bar { A } _ { i } } \bar { V } _ { i } = \bar { A } _ { i } \big ( P _ { i } ^ { \top } V W _ { i } ^ { V } \big ) ,
+$$
+
+so the full attention is now replaced with the implicit product of two low-rank matrices $\bar { A } _ { i } \in \mathbb { R } ^ { n \times r }$ and $P _ { i } ^ { \mathsf { T } } \in \mathbb { R } ^ { r \times n }$ , and the computational complexity is reduced to $O ( r n )$ . Note the effective attention weights of a query on all tokens still sum to 1. Our global attention allows each query to attend to all token embeddings within the same self-attention layer. In contrast, the sparse attention mechanisms [14, 16] need stack multiple layers to build such correlations.
+
+Application to Autoregressive Models: In autoregressive models, each token can only attend to the previous tokens, so the long-range attention should have a different range for different tokens. A straightforward way to implement our global attention is to update $\bar { K } _ { i } , \bar { V } _ { i }$ for each query recurrently, but this requires re-computing the projection in Eq. (3) for every token due to the nonlinearity of softmax, which results in $O ( r n ^ { 2 } )$ computational complexity. To preserve the linear complexity, for autoregressive models, we first divide the input sequence into equal-length segments with length $l$ , and apply our dynamic projection to extract $\bar { \bar { K } } _ { i } , \bar { \bar { V } } _ { i }$ from each segment. Each token can only attend to $\bar { K } _ { i } , \bar { V _ { i } }$ of segments that do not contain its future tokens. Formally, let $Q _ { t }$ be the query at position $t$ , $K _ { ( l - 1 ) s : l s } , V _ { ( l - 1 ) s : l s }$ be the key-value pairs from the $s$ -th segment, and $s _ { t } = \lfloor t / l \rfloor$ . For autoregressive models, we compute the long-range attention of $Q _ { t }$ by attending to $K _ { i , t } , V _ { i , t }$ , defined as
+
+$$
+\bar { K } _ { i , t } = [ P _ { i , 1 } ^ { \top } K _ { 1 : l } ; . . . ; P _ { i , s _ { t } } ^ { \top } K _ { ( l - 1 ) s _ { t } : l s _ { t } } ] W _ { i } ^ { K } , \bar { V } _ { i , t } = [ P _ { i , 1 } ^ { \top } V _ { 1 : l } ; . . . ; P _ { i , s _ { t } } ^ { \top } V _ { ( l - 1 ) s _ { t } : l s _ { t } } ] W _ { i } ^ { V } .
+$$
+
+In this way, the dynamic low-rank projection is applied to each segment only once in parallel, preserving the linear complexity and the high training speed. By comparison, Random Feature Attention [30] is slow at training due to the requirement for recurrence.
+
+# 3.4 Aggregating Long-range and Short-term Attentions
+
+To aggregate the local and long-range attentions, instead of adopting different attention mechanisms for different heads [12, 14, 34], we let each query at $i$ -th head attend to the union of keys and values from the local window and global low-rank projections, thus it can learn to select important information from either of them. We find this aggregation strategy works better than separating the heads in our initial trials with the autoregressive language models. Specifically, for the $i$ -th head, we denote the global low-rank projected keys and values as $\bar { K } _ { i } , \bar { V } _ { i } \in \dot { \mathbb { R } ^ { r \times d _ { k } } }$ , and the local keys and values as $\tilde { K } _ { t } , \tilde { \tilde { V } } _ { t } \in \mathbb { R } ^ { 2 w \times d }$ within the local window of position $t$ for the query $Q _ { t }$ . Then the $i$ -th attention $H _ { i , t }$ at position $t$ is
+
+$$
+H _ { i , t } = \mathrm { s o f t m a x } \left[ \frac { Q _ { t } W _ { i } ^ { Q } \left[ \tilde { K } _ { t } W _ { i } ^ { K } ; \bar { K } _ { i } \right] ^ { \mathsf { T } } } { \sqrt { d _ { k } } } \right] [ \tilde { V } _ { t } W _ { i } ^ { V } ; \bar { V } _ { i } ] .
+$$
+
+where $\left[ \cdot ; \cdot \right]$ denotes concatenating the matrices along the first dimension. Furthermore, we find a scale mismatch between the initial norms of $\tilde { K } _ { t } W _ { i } ^ { K }$ and ${ \bar { K } } _ { i }$ , which biases the attention to the local window at initialization for both language and vision tasks. We introduce a normalization strategy (DualLN) to align the norms and improve the effectiveness of the aggregation in the following.
+
+![](images/f18b82a395374db947c61de1d4db34317ba404018776556822de750fb1f9ea62.jpg)  
+Figure 2: Left: Ratios of the average $\ell _ { 2 }$ norms of the local window to global low-rank key/value embeddings at initialization. Without DualLN, the sparse and low-rank embeddings have a magnitude mismatch. With DualLN, the ratios will be 1.0 at every layer, which will facilitate optimization. Right: The validation loss of Transformer-LS with and without DualLN on enwik8 and text8.
+
+DualLN: For Transformers with Layer Normalization (LN) (see [44] for an illustration), the $K _ { i } , V _ { i }$ embeddings are the outputs of LN layers, so they have zero mean and unit variance at initialization. The $\ell _ { 2 }$ norm of vectors with zero-mean entries is proportional to their variance in expectation. We note a weighted average will reduce the variance and therefore the norm of such zero-mean vectors. As a result, the embedding vectors from low-rank attention in the weighted average $\bar { K } _ { i } , \bar { V } _ { i }$ of Eq. (3) will have smaller norms than the regular key and value embeddings from sliding window attention (see Figure 2 Left for an illustration). This scale mismatch causes two side effects. First, the inner product $Q _ { t } W _ { i } ^ { Q } { \bar { K } } _ { i } ^ { \mathsf { T } }$ from local-rank component tends to have smaller magnitude than the local window one, thus the attention scores on long-range attention is systematically smaller. Second, the key-value pairs $\bar { K } _ { i } , \bar { V } _ { i }$ for the low-rank attention will naturally have less impact on the direction of $H _ { i }$ even when low-rank and local window are assigned with same attention scores, since $\bar { V } _ { i }$ has smaller norms. Both effects lead to small gradients on the low-rank components and hinders the model from learning to effectively use the long-range correlations.
+
+To avoid such issues, we add two sets of Layer Normalizations after the key and value projections for the local window and global low-rank attentions, so that their scales are aligned at initialization, but the network can still learn to re-weight the norms after training. Specifically, the aggregated attention is now computed as
+
+$$
+{ \cal H } _ { i , t } = \mathrm { s o f t m a x } \left[ \frac { { Q _ { t } W _ { i } ^ { Q } \left[ { \cal L N } _ { L } ( { { \tilde { K } } _ { t } } W _ { i } ^ { K } ) ; { \cal L N } _ { G } ( { { \bar { K } } _ { i } } ) \right] ^ { \top } } } { \sqrt { d _ { k } } } \right] [ { \cal L N } _ { L } ( { { \tilde { V } } _ { t } } W _ { i } ^ { V } ) ; { \cal L N } _ { G } ( { { \bar { V } } _ { i } } ) ] ,
+$$
+
+where $\mathrm { L N } _ { L } ( \cdot ) , \mathrm { L N } _ { G } ( \cdot )$ denote the Layer Normalizations for the local and global attentions respectively. In practice, to maintain the consistency between the local attention and dynamic projection, we use $\mathsf { L } \bar { \mathsf { N } } _ { L } ( K ) , \mathsf { L } \mathsf { N } _ { L } ( V )$ instead of $K , V$ to compute $\bar { K } _ { i } , \bar { V } _ { i }$ in Eq. 3. As illustrated in Figure 2 Right, the Transformer-LS models trained with DualLN has consistently lower validation loss than the models without DualLN.
+
+# 4 Experiments
+
+In this section, we demonstrate the effectiveness and efficiency of our method in both language and vision domains. We use PyTorch for implementation and count the FLOPs using fvcore [45].
+
+# 4.1 Bidirectional Modeling on Long Range Arena and IMDb
+
+To evaluate Long-Short Transformer as a bidirectional encoder for long text, we train our models on the three NLP tasks, ListOps, Text, and Retrieval, from the recently proposed Long Range Arena (LRA) benchmark [20], following the setting of Peng et al. [30] and Tay et al. [46]. For fair comparisons, we use the PyTorch implementation and the same data preprocessing/split, training hyperparameters and model size from [18], except for Retrieval where we accidentally used more warmup steps and improved the results for all models. See Appendix B for more details. The results on these three tasks are given in Table 1. Results of the other two image-based tasks of LRA, as well as models implemented in JAX, are given in Appendix C and C.2.
+
+Table 1: Accuracy $( \% )$ and FLOPs (G) on Long Range Arena (LRA), with the model configs annotated (see Table 7 for more). All results are averages of 4 runs with different random seeds.   
+
+<table><tr><td>Task (mean ± std.) of sequence length</td><td colspan="2">ListOps (888±339)</td><td colspan="2">Text (1296 ± 893)</td><td colspan="2">Retrieval (3987 ± 560)</td><td>Average</td></tr><tr><td>Model</td><td>Acc.</td><td>FLOPs</td><td>Acc.</td><td>FLOPs</td><td>Acc.</td><td>FLOPs</td><td>Acc.</td></tr><tr><td>Full Attention [1]</td><td>37.13</td><td>1.21</td><td>65.35</td><td>4.57</td><td>82.30</td><td>9.14</td><td>61.59</td></tr><tr><td>Reformer [31] (2)</td><td>36.44</td><td>0.27</td><td>64.88</td><td>0.58</td><td>78.64</td><td>1.15</td><td>59.99</td></tr><tr><td>Linformer [17] (k=256)</td><td>37.38</td><td>0.41</td><td>56.12</td><td>0.81</td><td>79.37</td><td>1.62</td><td>57.62</td></tr><tr><td>Performer [28] (r = 256)</td><td>32.78</td><td>0.41</td><td>65.21</td><td>0.82</td><td>81.70</td><td>1.63</td><td>59.90</td></tr><tr><td>Nystromformer[18](l =128)</td><td>37.34</td><td>0.61</td><td>65.75</td><td>1.02</td><td>81.29</td><td>2.03</td><td>61.46</td></tr><tr><td>Transformer-LS (w,r = 8,32)</td><td>37.50</td><td>0.20</td><td>66.01</td><td>0.40</td><td>81.79</td><td>0.80</td><td>61.77</td></tr><tr><td>Dynamic Projection (best)</td><td>37.79</td><td>0.15</td><td>66.28</td><td>0.69</td><td>81.86</td><td>2.17</td><td>61.98</td></tr><tr><td>Transformer-LS (best)</td><td>38.36</td><td>0.16</td><td>68.40</td><td>0.29</td><td>81.95</td><td>2.17</td><td>62.90</td></tr></table>
+
+Table 2: Comparing the robustness of the models under test-time insertions and deletions. DP refers to long-range attention via Dynamic Projection, and Win. refers to sliding window attention.   
+
+<table><tr><td rowspan="2">Task Test Perturb</td><td colspan="3">Text</td><td colspan="3">Retrieval</td></tr><tr><td>None</td><td>Insertion</td><td>Deletion</td><td>None</td><td>Insertion</td><td>Deletion</td></tr><tr><td>Linformer</td><td>56.12</td><td>55.94</td><td>54.91</td><td>79.37</td><td>53.66</td><td>51.75</td></tr><tr><td>DP</td><td>66.28</td><td>63.16</td><td>58.95</td><td>81.86</td><td>70.01</td><td>64.98</td></tr><tr><td>Linformer+ Win.</td><td>59.63</td><td>56.69</td><td>56.29</td><td>79.68</td><td>52.83</td><td>52.13</td></tr><tr><td>DP + Win. (ours)</td><td>68.40</td><td>66.34</td><td>62.62</td><td>81.95</td><td>69.93</td><td>64.19</td></tr></table>
+
+Table 3: Comparing the results of pretrained language models fine-tuned on IMDb.   
+
+<table><tr><td>Model</td><td>RoBERTa-base RoBERTa-large Longformer-base LS-base LS-large</td><td></td><td></td><td></td><td></td></tr><tr><td> Accuracy</td><td>95.3</td><td>96.5</td><td>95.7</td><td>96.0</td><td>96.8</td></tr></table>
+
+In addition, we follow the pretraining procedure of Longformer [14] to pretrain our models based on RoBERTa-base and RoBERTa-large [47], and fine-tune it on the IMDb sentiment classification dataset. The results are given in Table 3.
+
+Results. From Table 3, our base model outperforms Longformer-base, and our large model achieves improvements over RoBERTa-large, demonstrating the benefits of learning to model long sequences. Comparisons with models on LRA are given in Table 1. Transformer-LS (best) with the best configurations of $w , r$ for each task are given in Table 7 in Appendix B. We also report the results of using fixed hyperparameter $w = 8 , r = 3 2$ on all tasks. Overall, our Transformer-LS (best) is significantly better than other efficient Transformers, and the model with $w , r = 8 , 3 2$ performs favorably while using only about $50 \%$ to $70 \%$ computation compared to other efficient Transformers on all three tasks. The advantage of aggregating local and long-range attentions is the most significant on ListOps, which requires the model to understand the tree structures involving both long-term and short-term relations. On Retrieval, where document-level encoding capability is tested, we find our global attention more effective than window attention. The test accuracy of using only dynamic projection is about $10 \%$ higher than Linformer on Text (i.e., 66.28 vs. 56.12), which has the highest variance in sequence length (i.e. standard deviation 893). This demonstrates the improved flexibility of dynamic projection at learning representations for data with high variance in sequence length, compared to the learned but fixed projection of Linformer. Similarly, Linformer, Nyströmformer and our model outperform full attention on ListOps, indicating they may have better inductive bias, and efficient Transformers can have better efficacy beyond efficiency.
+
+Robustness of Dynamic Projection. In Table 2, we compare the robustness of Linformer and the proposed Dynamic Projection (DP) against insertion and deletion on Text and Retrieval tasks of LRA. We train the models on the original, clean training sets and only perturb their test sets. For insertion, we insert 10 random punctuations at 10 random locations of each test sample. For deletion, we delete all punctuations from the test samples. Both transforms are label-preserving in most cases. By design, dynamic projection is more robust against location changes.
+
+![](images/4e35688918560b95df1faea3ced37447c973045ec764520f5c395363eee27938.jpg)  
+Forward-backprop Time per Iteration (Autoregressive LM)
+
+![](images/a333c6dfca964b5bcfe654e6654b8084e932baa4210e0e064ae9391c72990b47.jpg)  
+Figure 3: Running time and memory consumption of Transformer-XL (full attention) and our TransformerLS on Char-LM. We increase the sequence length until we use up the 32GB of memory on a V100 GPU. Transformer-LS is the same smaller model in Table 4. We use dashed lines to represent the full attention Transformer and solid lines to represent our model. We use different colors to represent different batch sizes.
+
+Table 4: BPC (↓) of smaller models on enwik8 and text8 (left), and larger models on enwik8 (right).   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">#Param</td><td colspan="3">text8</td></tr><tr><td>Dev Test</td><td>enwik8 Dev</td><td>Test</td></tr><tr><td>T12 [49] Transformer-XL [9]</td><td>44M</td><td>1 1.18</td><td>1</td><td>1.11</td></tr><tr><td>Reformer [31]</td><td>41M -</td><td>1 1 - 1</td><td>1 1</td><td>1.06 1.05</td></tr><tr><td>Adaptive [50]</td><td>38M</td><td>1.05 1.11</td><td>1.04</td><td>1.02</td></tr><tr><td>BP-Transformer [51] Longformer [20]</td><td>38M 41M</td><td>1 1.11 1.04 1.10</td><td>1 1.02</td><td>1.02 1.00</td></tr></table>
+
+<table><tr><td colspan="3">Method #Param‘ TestBPC</td></tr><tr><td>Transformer-XL [9]</td><td>88M</td><td>1.03 0.99</td></tr><tr><td>Transformer-XL [9] Routing [32]</td><td>277M 223M</td><td>0.99</td></tr><tr><td>Longformer [14]</td><td>102M</td><td>0.99</td></tr><tr><td>Sparse [12]</td><td>95M</td><td>0.99</td></tr><tr><td>Adaptive [50]</td><td>209M</td><td>0.98</td></tr><tr><td>Compressive [10]</td><td>227M</td><td>0.97</td></tr><tr><td>Transformer-LS</td><td>110M</td><td>0.97</td></tr></table>
+
+# 4.2 Autoregressive Language Modeling
+
+We compare our method with other efficient transformers on the character-level language modeling where each input token is a character.
+
+Setup. We train and evaluate our model on enwik8 and text8, each with 100M characters and are divided into 90M, 5M, 5M for train, dev, test, following [48]. Our smaller 12-layer and larger 30-layer models are Pre-LN Transformers with the same width and depth as Longformer [20], except that we add relative position encoding to the projected segments in each layer. We adopt the cache mechanism of Transformer-XL [9], setting the cache size to be the same as the input sequence length. We follow similar training schedule as Longformer, and train our model in 3 phases with increasing sequence lengths. The input sequence lengths are 2048, 4096 and 8192 respectively for the 3 phases. By comparison, Longformer trains their model in 5 phases on GPUs with 48GB memory (The maximal of ours is 32GB) where the sequence length is 23,040 in the last phase. The window size of Longformer increases with depth and its average window size is 4352 in phase 5, while our effective number of attended tokens is 1280 on average in the last phase. Each experiment takes around 8 days to finish on 8 V100 GPUs. Detailed hyperparameters are shown in Appendix D. For testing, same as Longformer, we split the dataset into overlapping sequences of length 32K at a step size of 512, and evaluate the BPCs for predicting the next 512 tokens given the previous 32K characters.
+
+Results Table 4 shows comparisons on text8 and enwik8. Our method has achieved state-of-the-art results. On text8, we achieve a test BPC of 1.09 with the smaller model. On enwik8, our smaller model achieves a test BPC of 0.99, and outperforms the state-of-the-art models with comparable number of parameters. Our larger model obtains a test BPC of 0.97, on par with the Compressive Transformer with $2 \times$ parameters. Our results are consistently better than Longformer which is trained on longer sequences with 5 stages and 48 GPU memory. In Figure 3, we show our model is much more memory and computational efficient than full attention.
+
+Table 5: Test accuracies on ImageNet, ImageNet Real [52], and ImageNet V2 [53] of models trained on ImageNet-1K. Grey-colored rows are our results. ${ \mathrm { C v T ^ { * } } }$ -LS denotes our long-short term attention based on the non-official CvT implementation. ViL models with LS suffixes are our long-short term attention based on the official ViL implementation with relative positional bias. We also provide the latency of models tested using batch size 32 on the same V100 GPU. Our improvements over ViL is mainly from a better implementation of the short-term attention.   
+
+<table><tr><td>Model</td><td>(M)</td><td>Size</td><td>#Param Image FLOPs ImageNet (G)</td><td>top-1 (%)</td><td>Real top-1 (%) top-1 (%)</td><td>V2</td><td>Latency (s）</td></tr><tr><td>ResNet-50</td><td>25</td><td>224²</td><td>4.1</td><td>76.2</td><td>82.5</td><td>63.3</td><td>=</td></tr><tr><td>ResNet-101</td><td>45</td><td>2242</td><td>7.9</td><td>77.4</td><td>83.7</td><td>65.7</td><td></td></tr><tr><td>ResNet-152</td><td>60</td><td>2242</td><td>11</td><td>78.3</td><td>84.1</td><td>67.0</td><td>=</td></tr><tr><td>DeiT-S [36]</td><td>22</td><td>224²</td><td>4.6</td><td>79.8</td><td>85.7</td><td>68.5</td><td></td></tr><tr><td>DeiT-B [36]</td><td>86</td><td>2242</td><td>17.6</td><td>81.8</td><td>86.7</td><td>70.9</td><td></td></tr><tr><td>PVT-Medium [5]</td><td>44</td><td>2242</td><td>6.7</td><td>81.2</td><td>1</td><td>1</td><td></td></tr><tr><td>PVT-Large [5]</td><td>61</td><td>2242</td><td>9.8</td><td>81.7</td><td></td><td></td><td></td></tr><tr><td>Swin-S [38]</td><td>50</td><td>2242</td><td>8.7</td><td>83.2</td><td></td><td></td><td></td></tr><tr><td>Swin-B[38]</td><td>88</td><td>2242</td><td>15.4</td><td>83.5</td><td></td><td></td><td>0.115</td></tr><tr><td>PVTv2-B4 [54]</td><td>62.6</td><td>2242</td><td>10.1</td><td>83.6</td><td></td><td></td><td></td></tr><tr><td>PVTv2-B5 [54]</td><td>82.0</td><td>2242</td><td>11.8</td><td>83.8</td><td></td><td></td><td></td></tr><tr><td>ViT-B/16 [4]</td><td>86</td><td>3842</td><td>55.5</td><td>77.9</td><td></td><td></td><td></td></tr><tr><td>ViT-L/16 [4]</td><td>307</td><td>3842</td><td>191.1</td><td>76.5</td><td></td><td></td><td></td></tr><tr><td>DeiT-B [36]</td><td>86</td><td>3842</td><td>55.5</td><td>83.1</td><td></td><td></td><td></td></tr><tr><td>Swin-B [38]</td><td>88</td><td>3842</td><td>47.1</td><td>84.5</td><td>1</td><td>1</td><td>0.378</td></tr><tr><td>CvT-13 [6]</td><td>20</td><td>2242</td><td>6.7</td><td>81.6</td><td>86.7</td><td>70.4</td><td>0.122</td></tr><tr><td>CvT-21 [6]</td><td>32</td><td>2242</td><td>10.1</td><td>82.5</td><td>87.2</td><td>71.3</td><td>0.165</td></tr><tr><td>CvT*-LS-13</td><td>20.3</td><td>2242</td><td>4.9</td><td>81.9</td><td>87.0</td><td>70.5</td><td>0.083</td></tr><tr><td>CvT*-LS-17</td><td>23.7</td><td>2242</td><td>9.8</td><td>82.5</td><td>87.2</td><td>71.6</td><td>-</td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>224²</td><td>7.9</td><td>82.7</td><td>87.5</td><td>71.9</td><td>0.122</td></tr><tr><td>CvT*-LS-21S</td><td>30.1</td><td>2242</td><td>11.3</td><td>82.9</td><td>87.4</td><td>71.7</td><td>-</td></tr><tr><td>CvT-13 [6]</td><td>20</td><td>3842</td><td>31.9</td><td>83.0</td><td>87.9</td><td>71.9</td><td>1</td></tr><tr><td>CvT-21 [6]</td><td>32</td><td>3842</td><td>45.0</td><td>83.3</td><td>87.7</td><td>71.9</td><td></td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>3842</td><td>23.9</td><td>83.2</td><td>88.0</td><td>72.5</td><td></td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>4482</td><td>34.2</td><td>83.6</td><td>88.2</td><td>72.9</td><td></td></tr><tr><td>ViL-Small [14]</td><td>24.6</td><td>2242</td><td>4.9</td><td>82.4</td><td>-</td><td>1</td><td></td></tr><tr><td>ViL-Medium [14]</td><td>39.7</td><td>2242</td><td>8.7</td><td>83.5</td><td></td><td></td><td>0.106</td></tr><tr><td>ViL-Base [14]</td><td>55.7</td><td>2242</td><td>13.4</td><td>83.7</td><td></td><td></td><td>0.164</td></tr><tr><td>ViL-LS-Medium</td><td>39.8</td><td>2242</td><td>8.7</td><td>83.8</td><td></td><td></td><td>0.075</td></tr><tr><td>ViL-LS-Base</td><td>55.8</td><td>224²</td><td>13.4</td><td>84.1</td><td></td><td></td><td>0.113</td></tr><tr><td> ViL-LS-Medium</td><td>39.9</td><td>3842</td><td>28.7</td><td>84.4</td><td></td><td></td><td>0.271</td></tr></table>
+
+# 4.3 ImageNet Classification
+
+We train and evaluate the models on ImageNet-1K with $1 . 3 \mathbf { M }$ images and 1K classes. We use CvT [6] and ViL [11], state-of-the art vision transformer architectures, as the backbones and replace their attention mechanisms with our long-short term attention, denoted as ${ \mathrm { C v T ^ { * } } }$ -LS and ViL-size-LS in Table 5. CvT uses overlapping convolutions to extract dense patch embeddings from the input images and feature maps, resulting in a long sequence length in the early stages (e.g., $5 6 \times 5 6 = 3 1 3 6$ patches for images with $2 2 4 ^ { 2 }$ pixels). For ViL, our sliding window uses the same group size $w$ , but each token attends to at most $2 w \times 2 w$ (rounding when necessary) tokens inside the window, instead of $3 w \times 3 w$ as ViL, which allows adding our dynamic projection without increasing the FLOPs. We set $r = 8$ for the dynamic projections for both ViL-LS-Medium and ViL-LS-Base. Note that, our efficient attention mechanism does not depend on the particular architecture, and it can be applied to other vision transformers [e.g., 4, 36, 5]. Please refer to Appendix E for more details.
+
+Classification Results. The results are shown in the Table 5, where we also list test accuracies on ImageNet Real and ImageNet V2. Except for CvT, we compare with the original ViT [4] and the enhanced DeiT [36], PVT [5] that also uses multi-scale stragey, ViL [11] that uses window attention and global tokens to improve the efficiency. Training at high-resolution usually improves the test accuracy of vision transformer. With our long-short term attention, we can easily scale the training to higher resolution, and the performance of ${ \mathrm { C v T ^ { * } } }$ -LS and ViL-LS also improves. Our best model with CvT ${ \mathrm { C v T ^ { * } } }$ -LS-21 at $4 4 { \bar { 8 } } ^ { 2 }$ ) achieves $0 . 3 \%$ higher accuracy than the best reported result of CvT while using the same amount of parameters and $76 \%$ of its FLOPs. In CvT architecture, the spatial dimension of feature maps in earlier stages are large, representing more fine-grained details of the image. Similar to training with high-resolution images, the model should also benefit from denser feature maps. With our efficient long-short term attention, we can better utilize these fine-grained feature maps with less concerns about the computational budget. In this way, our ${ \mathrm { C v T ^ { * } } }$ -LS-17 achieves better result than CvT-21 at resolution 224 using fewer parameters and FLOPs, and our ${ \mathrm { C v T ^ { * } } }$ -LS-21S model further improves our ${ \mathrm { C v T ^ { * } } }$ -LS-21 model.
+
+Table 6: Robustness evaluation on various ImageNet datasets. Top-1/Acc.: Top-1 accuracy. mCE: Mean Corrupution Error. Mixed-same/Mixed-rand: accuracies on MIXED-SAME/MIXED-RAND subsets.   
+
+<table><tr><td>Model</td><td colspan="5">[Params|ImageNet|IN-C [56]|IN-A [57]|IN-R [58]]</td><td colspan="2">ImageNet-9 [59]</td></tr><tr><td></td><td>(M)</td><td>Top-1</td><td>mCE(↓）</td><td>Acc.</td><td>Acc.</td><td>|Mixed-same Mixed-rand</td><td></td></tr><tr><td>ResNet-50 [35]]</td><td>25.6</td><td>76.2</td><td>78.9</td><td>6.2</td><td>35.3</td><td>87.1</td><td>81.6</td></tr><tr><td>DeiT-S [36]</td><td>22.1</td><td>79.8</td><td>57.1</td><td>19.0</td><td>41.9</td><td>89.1</td><td>84.2</td></tr><tr><td>CvT-13</td><td>20</td><td>81.6</td><td>59.6</td><td>25.4</td><td>42.9</td><td>90.5</td><td>85.7</td></tr><tr><td>CvT-21</td><td>32</td><td>82.5</td><td>56.2</td><td>31.1</td><td>42.6</td><td>90.5</td><td>85.0</td></tr><tr><td>CvT*-LS-13</td><td>20.3</td><td>81.9</td><td>58.7</td><td>27.0</td><td>42.6</td><td>90.7</td><td>85.6</td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>82.7</td><td>55.2</td><td>29.3</td><td>45.0</td><td>91.5</td><td>85.8</td></tr></table>
+
+Our ViL-LS-Medium and ViL-LS-Base with long-short term attention improve the accuracies of ViL-Medium and ViL-Base from 83.5 and 83.7 to 83.8 and 84.1 respectively, without an increase in FLOPs. When increasing the resolution for training ViL-LS-Medium from $\dot { 2 } 2 4 ^ { 2 }$ to $3 8 4 ^ { 2 }$ , the FLOPs increased (approximately) linearly and the accuracy improved by $0 . 6 \%$ , showing our method still benefits greatly from increased resolution while maintaining the linear complexity in practice.
+
+Short-term Attention Suppresses Oversmoothing. By restricting tokens from different segments to attend to different windows, our short-term sparse local attention encourages diversity of the feature representations and helps to alleviate the over-smoothing problem [55] (where all queries extract similar information in deeper layers and the attention mechanism is less important), thus can fully utilize the depth of the network. As in [55], we provide the cosine similarity of patch embeddings of our ${ \mathrm { C v T ^ { * } } }$ -LS-13 and re-implemented CvT-13 (81.1 accuracy) in Figure 6 within Appendix. This is one of the reasons why our efficient attention mechanism can get even better results than the full attention CvT model in the same setting.
+
+Robustness evaluation on Diverse ImageNet Datasets. As vision models have been widely used in safety-critical applications (e.g. autonomous driving), their robustness is vital. In addition to out-of-distribution robustness (ImageNet-Real and Imageet-v2), we further investigate the robustness of our vision transformer against common corruption (ImageNet-C), semantic shifts (ImageNet-R), Background dependence (ImageNet-9) and natural adversarial examples (ImageNet-A). We compare our methods with standard classification methods, including CNN-based model (ResNet [35]) and Transformer-based models (DeiT [36]) with similar numbers of parameters. As shown in Table 6, we observe that our method significantly outperforms the CNN-based method (ResNet-50). Compared to DeiT, our models also achieve favorable improvements. These results indicate that the design of different attention mechanisms plays an important role for model robustness, which sheds new light on the design of robust vision transformers. More details and results can be found in Appendix E.
+
+# 5 Conclusion
+
+In this paper, we introduced Long-Short Transformer, an efficient transformer for long sequence modeling for both language and vision domain, including both bidirectional and autoregressive models. We design a novel global attention mechanism with linear computational and memory complexity in sequence length based on a dynamic projection. We identify the scale mismatch issue and propose the DualLN technique to eliminate the mismatch at initialization and more effectively aggregate the local and global attentions. We demonstrate that our method obtains the state-of-the-art results on the Long Range Arena, char-level language modeling and ImageNet classification. We look forward to extending our methods to more domains, including document QA, object detection and semantic segmentation on high-resolution images.
+
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parse/train/M_lkFOwVdYc/M_lkFOwVdYc_content_list.json

@@ -0,0 +1,1049 @@
+[
+    {
+        "type": "text",
+        "text": "Long-Short Transformer: Efficient Transformers for Language and Vision ",
+        "text_level": 1,
+        "bbox": [
+            202,
+            122,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chen $Z \\mathrm { { h u } ^ { \\ddag 1 * } }$ , Wei Ping†2, Chaowei Xiao†2,3, Mohammad Shoeybi2, Tom Goldstein1, Anima Anandkumar2,4, and Bryan Catanzaro2 ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1University of Maryland, College Park 2 NVIDIA 3Arizona State University 4California Institute of Technology ‡chenzhu@cs.umd.edu, †{wping, chaoweix}@nvidia.com ",
+        "bbox": [
+            251,
+            263,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            347,
+            535,
+            363
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers have achieved success in both language and vision domains. However, it is prohibitively expensive to scale them to long sequences such as long documents or high-resolution images, because self-attention mechanism has quadratic time and memory complexities with respect to the input sequence length. In this paper, we propose Long-Short Transformer (Transformer-LS), an efficient self-attention mechanism for modeling long sequences with linear complexity for both language and vision tasks. It aggregates a novel long-range attention with dynamic projection to model distant correlations and a short-term attention to capture fine-grained local correlations. We propose a dual normalization strategy to account for the scale mismatch between the two attention mechanisms. Transformer-LS can be applied to both autoregressive and bidirectional models without additional complexity. Our method outperforms the state-of-the-art models on multiple tasks in language and vision domains, including the Long Range Arena benchmark, autoregressive language modeling, and ImageNet classification. For instance, Transformer-LS achieves 0.97 test BPC on enwik8 using half the number of parameters than previous method, while being faster and is able to handle $3 \\times$ as long sequences compared to its full-attention version on the same hardware. On ImageNet, it can obtain the state-of-the-art results (e.g., a moderate size of $5 5 . 8 \\mathbf { M }$ model solely trained on $2 2 4 \\times 2 2 4$ ImageNet-1K can obtain Top-1 accuracy $8 4 . 1 \\%$ ), while being more scalable on high-resolution images. The source code and models are released at https://github.com/NVIDIA/transformer-ls. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformer-based models [1] have achieved great success in the domains of natural language processing (NLP) [2, 3] and computer vision [4–6]. These models benefit from the self-attention module, which can capture both adjacent and long-range correlations between tokens while efficiently scaling on modern hardware. However, the time and memory consumed by self-attention scale quadratically with the input length, making it very expensive to process long sequences. Many language and vision tasks benefit from modeling long sequences. In NLP, document-level tasks require processing long articles [e.g., 7, 8], and the performance of language models often increases with sequence length [e.g., 9, 10]. In computer vision, many tasks involve high-resolution images, which are converted to long sequences of image patches before being processed with Transformer models [4, 6, 11]. As a result, it is crucial to design an efficient attention mechanism for long sequence modeling that generalizes well across different domains. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Various methods have been proposed to reduce the quadratic cost of full attention. However, an efficient attention mechanism that generalizes well in both language and vision domains is less explored. One family of methods is to sparsify the attention matrix with predefined patterns such as sliding windows [e.g., 12–15] and random sparse patterns [16]. These methods leverage strong inductive biases to improve both computational and model performance, but they limit the capacity of a self-attention layer because each specific token can only attend to a subset of tokens. Another family of methods leverages low-rank projections to form a low resolution representation of the input sequence, but the successful application of these methods has been limited to certain NLP tasks [e.g., 17–19]. Unlike sparse attention, this family of methods allows each token to attend to the entire input sequence. However, due to the loss of high-fidelity token-wise information, their performance sometimes is not as good as full attention or sparse attention on tasks that require fine-grained local information, including standard benchmarks in language [20] and vision [21]. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite the rapid progress in efficient Transformers, some proposed architectures can only be applied to bidirectional models [e.g., 15, 16, 18]. Transformer-based autoregressive models have achieved great successes in language modeling [22], image synthesis [23], and text-to-image synthesis [24], which also involve long texts or high-resolution images. It is desirable to design an efficient transformer that can be applied to both autoregressive and bidirectional models. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we unify a local window attention and a novel long-range attention into a single efficient attention mechanism. We show that these two kinds of attention have complementary effects that together yield the state-of-the-art results on a range of tasks in language and vision, for both autoregressive and bidirectional models. Specifically, we make the following contributions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• We propose Long-Short Transformer (Transformer-LS), an efficient Transformer that integrates a dynamic projection based attention to model long-range correlations, and a local window attention to capture fine-grained correlations. Transformer-LS can be applied to both autoregressive and bidirectional models with linear time and memory complexity.   \n• We compute a dynamic low-rank projection, which depends on the content of the input sequence. In contrast to previous low-rank projection methods, our dynamic projection method is more flexible and robust to semantic-preserving positional variations (e.g., insertion, paraphrasing). We demonstrate that it outperforms previous low-rank methods [17, 18] on Long Range Arena benchmark [20].   \n• We identify a scale mismatch problem between the embeddings from the long-range and shortterm attentions, and design a simple but effective dual normalization strategy, termed DualLN, to account for the mismatch and enhance the effectiveness of the aggregation.   \n• We demonstrate that Long-Short Transformer, despite its low memory and runtime complexity, outperforms the state-of-the-art models on a set of tasks from Long Range Arena, and autoregressive language modeling on enwik8 and text8. In addition, the proposed efficient attention mechanism can be easily applied to the most recent vision transformer architectures [6, 11] and provides state-of-the-art results, while being more scalable to high-resolution images. We also investigate the robustness properties of the Transformer-LS on diverse ImageNet datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 Efficient Transformers ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In recent years, many methods have been introduced for dealing with the quadratic cost of full attention. In general, they can be categorized as follows: i) Sparse attention mechanism with predefined patterns (e.g., sliding window), including Sparse Transformer [12], Image Transformer [13], Axial Transformer [25] for modeling images, and Longformer [14], blockwise self-attention [26], ETC [15], Big Bird [16] for modeling language. ii) Low-rank projection attention, including Linformer [17], Nyströmformer [18], Synthesizer [19]. For example, Linformer uses linear layers to project the original high resolution keys $( K )$ and values $( V )$ with length $n$ to low resolution with size $r$ $( r \\ll n )$ ) and allows all query tokens $( Q )$ to attend these compressed representations. iii) Memory-based mechanisms like Compressive Transformer [10] and Set Transformer [27], which use extra memories for caching global long-range information for use in computing attention between distant tokens. iv) Kernel-based approximation of the attention matrix, including Performer [28], Linear ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/cac6035bb2eb7c5eb4e197f0f078ccb16d4a2c66c0c57d273bbe65a2ec761215.jpg",
+        "image_caption": [
+            "Figure 1: Long-short term attention of a single attention head. Here, the sequence length $n = 8$ , hidden dimension $d = 3$ , local window segment size $w = 2$ , and rank of dynamic projection $r = 3$ . Within the figure, $K ( V )$ denotes key $K$ or value $V$ . In the left figure, we virtually replicate $K$ or $V \\in \\mathbb { R } ^ { n \\times d }$ into $n$ rows, and highlight the keys and values within the attention span (denoted as $\\tilde { K } ( \\tilde { V } ) )$ of all $_ n$ queries $Q$ for the short-term attention. In the middle figure, all queries attend to the same projected keys $\\bar { K }$ and values $\\bar { V }$ within the long-term attention. In the right figure, $\\tilde { K } ( \\tilde { V } )$ and $\\bar { K } ( \\bar { V } )$ are first normalized with two sets of LayerNorms, and the queries attend to normalized $\\tilde { K } ( \\tilde { V } )$ and $\\bar { K } ( \\bar { V } )$ within their attention span simultaneously. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transformer [29], and Random Feature Attention [30]. vi) Similarity and clustering based methods, including Reformer [31], Routing Transformer [32], and Sinkhorn Transformer [33]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our method seamlessly integrates both low-rank projection and local window attentions, to leverage their strengths for modeling long-range and short-term correlations. In particular, our long-range attention uses a dynamic low-rank projection to encode the input sequence, and outperforms the previous low-rank projection method used by the Linformer [17]. In the similar vein, a few other methods also try to combine the strengths of different methods. For example, Longformer [14] and ETC [15] augment local window attention with task motivated global tokens. Such global tokens may not be applicable for some tasks (e.g., autoregressive modelling). BigBird [16] further combines local window and global token attention with random sparse attention. It is not applicable in autoregressive tasks because the global token and random sparse pattern are introduced. To compress the model footprint on edge devices, Lite Transformer [34] combines convolution and self-attention, but it still has quadratic complexity for long sequences. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Vision Transformers ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Vision Transformer (ViT) [4] splits images as small patches and treats the patches as the input word tokens. It uses a standard transformer for image classification and has shown to outperform convolutional neural networks (e.g., ResNet [35]) with sufficient training data. DeiT [36] has applied the teacher-student strategy to alleviate the data efficiency problem of ViT and has shown strong comparable performance using only the standard ImageNet dataset [37]. Instead of applying transformer at a single low resolution of patches (e.g., $1 6 \\times 1 6$ patches), very recent works, including Pyramid Vision Transformer (PVT) [5], Swin-Transformer [38], T2T-ViT [39], Vision Longformer (ViL) [11] and Convolutional Vision Transformer (CvT) [6], stack a pyramid of ViTs to form a multi-scale architecture and model long sequences of image patches at much higher resolution (e.g., $5 6 \\times 5 6 = 3 1 3 6$ patches for images with $2 2 4 \\times 2 2 4$ pixels). Most of these methods have quadratic complexity of self-attention with respect to the input image size. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To reduce the complexity, Swin-Transformer [38] achieves linear complexity by limiting the computation of self-attention only within each local window. HaloNet [40] applies local attention on blocked images and only has quadratic complexity with respect to the size of the block. Perceiver [41] uses cross-attention between data and latent arrays to replace the self-attention on data to remove the quadratic complexity bottleneck. Vision Longformer (ViL) [11], another concurrent work, achieves linear complexity by adapting Longformer [14] to Vision. ViL augments local window attention with task-specific global tokens, but the global tokens are not applicable for decoding task (e.g., image synthesis [23, 24]). In contrast, our method reduces the quadratic cost to linear cost by combining local window attention with global dynamic projection attention, which can be applied to both encoding and decoding tasks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Long-Short Transformer ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transformer-LS approximates the full attention by aggregating long-range and short-term attentions, while maintaining its ability to capture correlations between all input tokens. In this section, we first introduce the preliminaries of multi-head attention in Transformer. Then, we present the short-term attention via sliding window, and long-range attention via dynamic projection, respectively. After that, we propose the aggregating method and dual normalization (DualLN) strategy. See Figure 1 for an illustration of our long-short term attention. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.1 Preliminaries and Notations ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Multi-head attention is a core component of the Transformer [1], which computes contextual representations for each token by attending to the whole input sequence at different representation subspaces. It is defined as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b1425dd2845fe7c1790814f7554ac36ddd1ae3a72dc8320553e7b7f071cad928.jpg",
+        "text": "$$\n\\mathrm { M u l t i H e a d } ( Q , K , V ) = \\mathrm { C o n c a t } ( H _ { 1 } , H _ { 2 } , . . . , H _ { h } ) W ^ { O } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $Q , K , V \\in \\mathbb { R } ^ { n \\times d }$ are the query, key and value embeddings, $W ^ { O } \\in \\mathbb { R } ^ { d \\times d }$ is the projection matrix for output, the $i$ -th head $\\bar { H _ { i } } \\in \\bar { \\mathbb { R } } ^ { n \\times \\check { d } _ { k } }$ is the scaled dot-product attention, and $d _ { k } = d / h$ is the embedding dimension of each head, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1c0d01198277cab35c64eec76d0808133f18f655781edfcd64984d196719b333.jpg",
+        "text": "$$\nH _ { i } = \\mathrm { A t t e n t i o n } ( Q W _ { i } ^ { Q } , K W _ { i } ^ { K } , V W _ { i } ^ { V } ) = \\mathrm { s o f t m a x } \\left[ \\frac { Q W _ { i } ^ { Q } \\left( K W _ { i } ^ { K } \\right) ^ { \\mathsf { T } } } { \\sqrt { d _ { k } } } \\right] V W _ { i } ^ { V } = A _ { i } V W _ { i } ^ { V } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            297,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { i } ^ { Q } , W _ { i } ^ { K } , W _ { i } ^ { V } \\in \\mathbb { R } ^ { d \\times d _ { k } }$ are learned projection matrices, and $A _ { i } \\in \\mathbb { R } ^ { n \\times n }$ denotes the full attention matrix for each attention head. The complexity of computing and storing $A _ { i }$ is $O ( n ^ { 2 } )$ , which can be prohibitive when $n$ is large. For simplicity, our discussion below is based on the case of 1D input sequences. It is straightforward to extend to the 2D image data given a predetermined order. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Short-term Attention via Segment-wise Sliding Window ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use the simple yet effective sliding window attention to capture fine-grained local correlations, where each query attends to nearby tokens within a fixed-size neighborhood. Similar techniques have also been adopted in [14, 16, 11]. Specifically, we divide the input sequence into disjoint segments with length $w$ for efficiency reason. All tokens within a segment attend to all tokens within its home segment, as well as $w / 2$ consecutive tokens on the left and right side of its home segment (zero-padding when necessary), resulting in an attention span over a total of $2 w$ key-value pairs. See Figure 5 in Appendix for an illustration. For each query $Q _ { t }$ at the position $t$ within the $i$ -th head, we denote the $2 w$ key-value pairs within its window as $\\bar { K } _ { t } , \\tilde { V } _ { t } \\in \\mathbb { R } ^ { \\bar { 2 w } \\times d }$ . For implementation with PyTorch, this segment-wise sliding window attention is faster than the per-token sliding window attention where each token attends to itself and $w$ tokens to its left and right, and its memory consumption scales linearly with sequence length; see [14] and our Figure 3 for more details. ",
+        "bbox": [
+            173,
+            440,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The sliding window attention can be augmented to capture long-range correlations in part, by introducing different dilations to different heads of sliding window attention [14]. However, the dilation configurations for different heads need further tuning and an efficient implementation of multi-head attention with different dilations is non-trivial. A more efficient alternative is to augment sliding window attention with random sparse attention [16], but this does not guarantee that the long-range correlations are captured in each layer as in full attention. In the following section, we propose our long-range attention to address this issue. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 Long-range Attention via Dynamic Projections ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            712,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Previous works have shown that the self-attention matrix can be well approximated by the product of low-rank matrices [17]. By replacing the full attention with the product of low-rank matrices [42, 19, 18, 43, 28], each query is able to attend to all tokens. Linformer [17] is one of the most representative models in this category. It learns a fixed projection matrix to reduce the length of the keys and values, but the fixed projection is inflexible to semantic-preserving positional variations. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Starting from these observations, we parameterize the dynamic low-rank projection at $i$ -th head as $P _ { i } = f ( K ) \\in \\mathbb { R } ^ { n \\times r }$ , where $r \\ll n$ is the low rank size and $P _ { i }$ depends on all the keys $K \\in \\mathbb { R } ^ { n \\times d }$ of input sequence. It projects the $( n \\times d _ { k } )$ -dimensional key embeddings $K W _ { i } ^ { K }$ and value embeddings $V W _ { i } ^ { V }$ into shorter, $( r \\times d _ { k } )$ -dimensional key ${ \\bar { K } } _ { i }$ and value $\\bar { V } _ { i }$ embeddings. Unlike Linformer [17], the low-rank projection matrix is dynamic, which depends on the input sequence and is intended to be more flexible and robust to, e.g., insertion, deletion, paraphrasing, and other operations that change sequence length. See Table 2 for examples. Note that, the query embeddings $Q \\bar { W } _ { i } ^ { Q } \\in \\mathbb { R } ^ { n \\times d _ { k } }$ are kept at the same length, and we let each query attend to ${ \\bar { K } } _ { i }$ and $\\bar { V } _ { i }$ . In this way, the full $( n \\times n )$ attention matrix can be decomposed into the product of two matrices with $r$ columns or rows. Specifically, we define the dynamic projection matrix $P _ { i } \\in \\mathbb { R } ^ { n \\times r }$ and the key-value embeddings $\\bar { K } _ { i } , \\dot { \\bar { V } } _ { i } \\in \\mathbb { R } ^ { r \\times \\mathbf { \\dot { d } } _ { k } }$ of low-rank attention as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d9f00fe1ec37f70c829dc50e309433636959eb36274b3916379c63e366034540.jpg",
+        "text": "$$\nP _ { i } = \\mathrm { s o f t m a x } ( K W _ { i } ^ { P } ) , \\bar { K } _ { i } = P _ { i } ^ { \\intercal } K W _ { i } ^ { K } , \\bar { V } _ { i } = P _ { i } ^ { \\intercal } V W _ { i } ^ { V } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { i } ^ { P } \\in \\mathbb { R } ^ { d \\times r }$ are learnable parameters,2 and the softmax normalizes the projection weights on the first dimension over all $n$ tokens, which stabilizes training in our experiments. Note that $K = V$ in all the experiments we have considered, so $P _ { i }$ remains the same if it depends on $V$ . The computational complexity of Eq. 3 is $O ( r n )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To see how the full attention is replaced by the product of low-rank matrices, we compute each head $H _ { i } \\in \\mathbb { R } ^ { n \\times d _ { k } }$ of long-range attention as, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/604a476c66b5dbfd8aa4ebcbe2d5d477bc999c8fba390515cf768f90299ba393.jpg",
+        "text": "$$\n\\bar { H } _ { i } = \\underbrace { \\mathrm { s o f t m a x } \\left[ \\frac { Q W _ { i } ^ { Q } \\bar { K } _ { i } ^ { \\top } } { \\sqrt { d _ { k } } } \\right] } _ { \\bar { A } _ { i } } \\bar { V } _ { i } = \\bar { A } _ { i } \\big ( P _ { i } ^ { \\top } V W _ { i } ^ { V } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "so the full attention is now replaced with the implicit product of two low-rank matrices $\\bar { A } _ { i } \\in \\mathbb { R } ^ { n \\times r }$ and $P _ { i } ^ { \\mathsf { T } } \\in \\mathbb { R } ^ { r \\times n }$ , and the computational complexity is reduced to $O ( r n )$ . Note the effective attention weights of a query on all tokens still sum to 1. Our global attention allows each query to attend to all token embeddings within the same self-attention layer. In contrast, the sparse attention mechanisms [14, 16] need stack multiple layers to build such correlations. ",
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+    {
+        "type": "text",
+        "text": "Application to Autoregressive Models: In autoregressive models, each token can only attend to the previous tokens, so the long-range attention should have a different range for different tokens. A straightforward way to implement our global attention is to update $\\bar { K } _ { i } , \\bar { V } _ { i }$ for each query recurrently, but this requires re-computing the projection in Eq. (3) for every token due to the nonlinearity of softmax, which results in $O ( r n ^ { 2 } )$ computational complexity. To preserve the linear complexity, for autoregressive models, we first divide the input sequence into equal-length segments with length $l$ , and apply our dynamic projection to extract $\\bar { \\bar { K } } _ { i } , \\bar { \\bar { V } } _ { i }$ from each segment. Each token can only attend to $\\bar { K } _ { i } , \\bar { V _ { i } }$ of segments that do not contain its future tokens. Formally, let $Q _ { t }$ be the query at position $t$ , $K _ { ( l - 1 ) s : l s } , V _ { ( l - 1 ) s : l s }$ be the key-value pairs from the $s$ -th segment, and $s _ { t } = \\lfloor t / l \\rfloor$ . For autoregressive models, we compute the long-range attention of $Q _ { t }$ by attending to $K _ { i , t } , V _ { i , t }$ , defined as ",
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+    {
+        "type": "equation",
+        "img_path": "images/2b8af8a2115394307ff82943ba6e7fc875b3af1ef6ab96d766c21315e07cc9fc.jpg",
+        "text": "$$\n\\bar { K } _ { i , t } = [ P _ { i , 1 } ^ { \\top } K _ { 1 : l } ; . . . ; P _ { i , s _ { t } } ^ { \\top } K _ { ( l - 1 ) s _ { t } : l s _ { t } } ] W _ { i } ^ { K } , \\bar { V } _ { i , t } = [ P _ { i , 1 } ^ { \\top } V _ { 1 : l } ; . . . ; P _ { i , s _ { t } } ^ { \\top } V _ { ( l - 1 ) s _ { t } : l s _ { t } } ] W _ { i } ^ { V } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In this way, the dynamic low-rank projection is applied to each segment only once in parallel, preserving the linear complexity and the high training speed. By comparison, Random Feature Attention [30] is slow at training due to the requirement for recurrence. ",
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+        "type": "text",
+        "text": "3.4 Aggregating Long-range and Short-term Attentions ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To aggregate the local and long-range attentions, instead of adopting different attention mechanisms for different heads [12, 14, 34], we let each query at $i$ -th head attend to the union of keys and values from the local window and global low-rank projections, thus it can learn to select important information from either of them. We find this aggregation strategy works better than separating the heads in our initial trials with the autoregressive language models. Specifically, for the $i$ -th head, we denote the global low-rank projected keys and values as $\\bar { K } _ { i } , \\bar { V } _ { i } \\in \\dot { \\mathbb { R } ^ { r \\times d _ { k } } }$ , and the local keys and values as $\\tilde { K } _ { t } , \\tilde { \\tilde { V } } _ { t } \\in \\mathbb { R } ^ { 2 w \\times d }$ within the local window of position $t$ for the query $Q _ { t }$ . Then the $i$ -th attention $H _ { i , t }$ at position $t$ is ",
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+        "type": "equation",
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+        "text": "$$\nH _ { i , t } = \\mathrm { s o f t m a x } \\left[ \\frac { Q _ { t } W _ { i } ^ { Q } \\left[ \\tilde { K } _ { t } W _ { i } ^ { K } ; \\bar { K } _ { i } \\right] ^ { \\mathsf { T } } } { \\sqrt { d _ { k } } } \\right] [ \\tilde { V } _ { t } W _ { i } ^ { V } ; \\bar { V } _ { i } ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\left[ \\cdot ; \\cdot \\right]$ denotes concatenating the matrices along the first dimension. Furthermore, we find a scale mismatch between the initial norms of $\\tilde { K } _ { t } W _ { i } ^ { K }$ and ${ \\bar { K } } _ { i }$ , which biases the attention to the local window at initialization for both language and vision tasks. We introduce a normalization strategy (DualLN) to align the norms and improve the effectiveness of the aggregation in the following. ",
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+        "type": "image",
+        "img_path": "images/f18b82a395374db947c61de1d4db34317ba404018776556822de750fb1f9ea62.jpg",
+        "image_caption": [
+            "Figure 2: Left: Ratios of the average $\\ell _ { 2 }$ norms of the local window to global low-rank key/value embeddings at initialization. Without DualLN, the sparse and low-rank embeddings have a magnitude mismatch. With DualLN, the ratios will be 1.0 at every layer, which will facilitate optimization. Right: The validation loss of Transformer-LS with and without DualLN on enwik8 and text8. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "DualLN: For Transformers with Layer Normalization (LN) (see [44] for an illustration), the $K _ { i } , V _ { i }$ embeddings are the outputs of LN layers, so they have zero mean and unit variance at initialization. The $\\ell _ { 2 }$ norm of vectors with zero-mean entries is proportional to their variance in expectation. We note a weighted average will reduce the variance and therefore the norm of such zero-mean vectors. As a result, the embedding vectors from low-rank attention in the weighted average $\\bar { K } _ { i } , \\bar { V } _ { i }$ of Eq. (3) will have smaller norms than the regular key and value embeddings from sliding window attention (see Figure 2 Left for an illustration). This scale mismatch causes two side effects. First, the inner product $Q _ { t } W _ { i } ^ { Q } { \\bar { K } } _ { i } ^ { \\mathsf { T } }$ from local-rank component tends to have smaller magnitude than the local window one, thus the attention scores on long-range attention is systematically smaller. Second, the key-value pairs $\\bar { K } _ { i } , \\bar { V } _ { i }$ for the low-rank attention will naturally have less impact on the direction of $H _ { i }$ even when low-rank and local window are assigned with same attention scores, since $\\bar { V } _ { i }$ has smaller norms. Both effects lead to small gradients on the low-rank components and hinders the model from learning to effectively use the long-range correlations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To avoid such issues, we add two sets of Layer Normalizations after the key and value projections for the local window and global low-rank attentions, so that their scales are aligned at initialization, but the network can still learn to re-weight the norms after training. Specifically, the aggregated attention is now computed as ",
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+        "img_path": "images/0db0be27695b5792b00fe9b20c94c93debe7d987c4c98883952cbab7d5c15739.jpg",
+        "text": "$$\n{ \\cal H } _ { i , t } = \\mathrm { s o f t m a x } \\left[ \\frac { { Q _ { t } W _ { i } ^ { Q } \\left[ { \\cal L N } _ { L } ( { { \\tilde { K } } _ { t } } W _ { i } ^ { K } ) ; { \\cal L N } _ { G } ( { { \\bar { K } } _ { i } } ) \\right] ^ { \\top } } } { \\sqrt { d _ { k } } } \\right] [ { \\cal L N } _ { L } ( { { \\tilde { V } } _ { t } } W _ { i } ^ { V } ) ; { \\cal L N } _ { G } ( { { \\bar { V } } _ { i } } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { L N } _ { L } ( \\cdot ) , \\mathrm { L N } _ { G } ( \\cdot )$ denote the Layer Normalizations for the local and global attentions respectively. In practice, to maintain the consistency between the local attention and dynamic projection, we use $\\mathsf { L } \\bar { \\mathsf { N } } _ { L } ( K ) , \\mathsf { L } \\mathsf { N } _ { L } ( V )$ instead of $K , V$ to compute $\\bar { K } _ { i } , \\bar { V } _ { i }$ in Eq. 3. As illustrated in Figure 2 Right, the Transformer-LS models trained with DualLN has consistently lower validation loss than the models without DualLN. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we demonstrate the effectiveness and efficiency of our method in both language and vision domains. We use PyTorch for implementation and count the FLOPs using fvcore [45]. ",
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+    {
+        "type": "text",
+        "text": "4.1 Bidirectional Modeling on Long Range Arena and IMDb ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To evaluate Long-Short Transformer as a bidirectional encoder for long text, we train our models on the three NLP tasks, ListOps, Text, and Retrieval, from the recently proposed Long Range Arena (LRA) benchmark [20], following the setting of Peng et al. [30] and Tay et al. [46]. For fair comparisons, we use the PyTorch implementation and the same data preprocessing/split, training hyperparameters and model size from [18], except for Retrieval where we accidentally used more warmup steps and improved the results for all models. See Appendix B for more details. The results on these three tasks are given in Table 1. Results of the other two image-based tasks of LRA, as well as models implemented in JAX, are given in Appendix C and C.2. ",
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+    {
+        "type": "table",
+        "img_path": "images/b417a2969e49d11a75554c450369708b7d3d2906a8e6d9d77b30007dcbb8e710.jpg",
+        "table_caption": [
+            "Table 1: Accuracy $( \\% )$ and FLOPs (G) on Long Range Arena (LRA), with the model configs annotated (see Table 7 for more). All results are averages of 4 runs with different random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task (mean ± std.) of sequence length</td><td colspan=\"2\">ListOps (888±339)</td><td colspan=\"2\">Text (1296 ± 893)</td><td colspan=\"2\">Retrieval (3987 ± 560)</td><td>Average</td></tr><tr><td>Model</td><td>Acc.</td><td>FLOPs</td><td>Acc.</td><td>FLOPs</td><td>Acc.</td><td>FLOPs</td><td>Acc.</td></tr><tr><td>Full Attention [1]</td><td>37.13</td><td>1.21</td><td>65.35</td><td>4.57</td><td>82.30</td><td>9.14</td><td>61.59</td></tr><tr><td>Reformer [31] (2)</td><td>36.44</td><td>0.27</td><td>64.88</td><td>0.58</td><td>78.64</td><td>1.15</td><td>59.99</td></tr><tr><td>Linformer [17] (k=256)</td><td>37.38</td><td>0.41</td><td>56.12</td><td>0.81</td><td>79.37</td><td>1.62</td><td>57.62</td></tr><tr><td>Performer [28] (r = 256)</td><td>32.78</td><td>0.41</td><td>65.21</td><td>0.82</td><td>81.70</td><td>1.63</td><td>59.90</td></tr><tr><td>Nystromformer[18](l =128)</td><td>37.34</td><td>0.61</td><td>65.75</td><td>1.02</td><td>81.29</td><td>2.03</td><td>61.46</td></tr><tr><td>Transformer-LS (w,r = 8,32)</td><td>37.50</td><td>0.20</td><td>66.01</td><td>0.40</td><td>81.79</td><td>0.80</td><td>61.77</td></tr><tr><td>Dynamic Projection (best)</td><td>37.79</td><td>0.15</td><td>66.28</td><td>0.69</td><td>81.86</td><td>2.17</td><td>61.98</td></tr><tr><td>Transformer-LS (best)</td><td>38.36</td><td>0.16</td><td>68.40</td><td>0.29</td><td>81.95</td><td>2.17</td><td>62.90</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e99bd67e90931e5d378b80f30a935c45019599e74660e944239ddf019f101eef.jpg",
+        "table_caption": [
+            "Table 2: Comparing the robustness of the models under test-time insertions and deletions. DP refers to long-range attention via Dynamic Projection, and Win. refers to sliding window attention. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task Test Perturb</td><td colspan=\"3\">Text</td><td colspan=\"3\">Retrieval</td></tr><tr><td>None</td><td>Insertion</td><td>Deletion</td><td>None</td><td>Insertion</td><td>Deletion</td></tr><tr><td>Linformer</td><td>56.12</td><td>55.94</td><td>54.91</td><td>79.37</td><td>53.66</td><td>51.75</td></tr><tr><td>DP</td><td>66.28</td><td>63.16</td><td>58.95</td><td>81.86</td><td>70.01</td><td>64.98</td></tr><tr><td>Linformer+ Win.</td><td>59.63</td><td>56.69</td><td>56.29</td><td>79.68</td><td>52.83</td><td>52.13</td></tr><tr><td>DP + Win. (ours)</td><td>68.40</td><td>66.34</td><td>62.62</td><td>81.95</td><td>69.93</td><td>64.19</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6ec81474b10f51f4b1cbf754c0abed789ebd23a137dfbcd6ded44ef1125b311e.jpg",
+        "table_caption": [
+            "Table 3: Comparing the results of pretrained language models fine-tuned on IMDb. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>RoBERTa-base RoBERTa-large Longformer-base LS-base LS-large</td><td></td><td></td><td></td><td></td></tr><tr><td> Accuracy</td><td>95.3</td><td>96.5</td><td>95.7</td><td>96.0</td><td>96.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "In addition, we follow the pretraining procedure of Longformer [14] to pretrain our models based on RoBERTa-base and RoBERTa-large [47], and fine-tune it on the IMDb sentiment classification dataset. The results are given in Table 3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. From Table 3, our base model outperforms Longformer-base, and our large model achieves improvements over RoBERTa-large, demonstrating the benefits of learning to model long sequences. Comparisons with models on LRA are given in Table 1. Transformer-LS (best) with the best configurations of $w , r$ for each task are given in Table 7 in Appendix B. We also report the results of using fixed hyperparameter $w = 8 , r = 3 2$ on all tasks. Overall, our Transformer-LS (best) is significantly better than other efficient Transformers, and the model with $w , r = 8 , 3 2$ performs favorably while using only about $50 \\%$ to $70 \\%$ computation compared to other efficient Transformers on all three tasks. The advantage of aggregating local and long-range attentions is the most significant on ListOps, which requires the model to understand the tree structures involving both long-term and short-term relations. On Retrieval, where document-level encoding capability is tested, we find our global attention more effective than window attention. The test accuracy of using only dynamic projection is about $10 \\%$ higher than Linformer on Text (i.e., 66.28 vs. 56.12), which has the highest variance in sequence length (i.e. standard deviation 893). This demonstrates the improved flexibility of dynamic projection at learning representations for data with high variance in sequence length, compared to the learned but fixed projection of Linformer. Similarly, Linformer, Nyströmformer and our model outperform full attention on ListOps, indicating they may have better inductive bias, and efficient Transformers can have better efficacy beyond efficiency. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Robustness of Dynamic Projection. In Table 2, we compare the robustness of Linformer and the proposed Dynamic Projection (DP) against insertion and deletion on Text and Retrieval tasks of LRA. We train the models on the original, clean training sets and only perturb their test sets. For insertion, we insert 10 random punctuations at 10 random locations of each test sample. For deletion, we delete all punctuations from the test samples. Both transforms are label-preserving in most cases. By design, dynamic projection is more robust against location changes. ",
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+    {
+        "type": "image",
+        "img_path": "images/4e35688918560b95df1faea3ced37447c973045ec764520f5c395363eee27938.jpg",
+        "image_caption": [
+            "Forward-backprop Time per Iteration (Autoregressive LM) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/a333c6dfca964b5bcfe654e6654b8084e932baa4210e0e064ae9391c72990b47.jpg",
+        "image_caption": [
+            "Figure 3: Running time and memory consumption of Transformer-XL (full attention) and our TransformerLS on Char-LM. We increase the sequence length until we use up the 32GB of memory on a V100 GPU. Transformer-LS is the same smaller model in Table 4. We use dashed lines to represent the full attention Transformer and solid lines to represent our model. We use different colors to represent different batch sizes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b15e3ab0433b33666b9b2af449b3cb7fa70cd6eb102901b2693ff58050f15211.jpg",
+        "table_caption": [
+            "Table 4: BPC (↓) of smaller models on enwik8 and text8 (left), and larger models on enwik8 (right). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">#Param</td><td colspan=\"3\">text8</td></tr><tr><td>Dev Test</td><td>enwik8 Dev</td><td>Test</td></tr><tr><td>T12 [49] Transformer-XL [9]</td><td>44M</td><td>1 1.18</td><td>1</td><td>1.11</td></tr><tr><td>Reformer [31]</td><td>41M -</td><td>1 1 - 1</td><td>1 1</td><td>1.06 1.05</td></tr><tr><td>Adaptive [50]</td><td>38M</td><td>1.05 1.11</td><td>1.04</td><td>1.02</td></tr><tr><td>BP-Transformer [51] Longformer [20]</td><td>38M 41M</td><td>1 1.11 1.04 1.10</td><td>1 1.02</td><td>1.02 1.00</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/a82c29f64795331a53468152cf14d4c8d87aca0d05b0b0e431c67ba64026a2f7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Method #Param‘ TestBPC</td></tr><tr><td>Transformer-XL [9]</td><td>88M</td><td>1.03 0.99</td></tr><tr><td>Transformer-XL [9] Routing [32]</td><td>277M 223M</td><td>0.99</td></tr><tr><td>Longformer [14]</td><td>102M</td><td>0.99</td></tr><tr><td>Sparse [12]</td><td>95M</td><td>0.99</td></tr><tr><td>Adaptive [50]</td><td>209M</td><td>0.98</td></tr><tr><td>Compressive [10]</td><td>227M</td><td>0.97</td></tr><tr><td>Transformer-LS</td><td>110M</td><td>0.97</td></tr></table>",
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+        "type": "text",
+        "text": "4.2 Autoregressive Language Modeling ",
+        "text_level": 1,
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+        "text": "We compare our method with other efficient transformers on the character-level language modeling where each input token is a character. ",
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+        "text": "Setup. We train and evaluate our model on enwik8 and text8, each with 100M characters and are divided into 90M, 5M, 5M for train, dev, test, following [48]. Our smaller 12-layer and larger 30-layer models are Pre-LN Transformers with the same width and depth as Longformer [20], except that we add relative position encoding to the projected segments in each layer. We adopt the cache mechanism of Transformer-XL [9], setting the cache size to be the same as the input sequence length. We follow similar training schedule as Longformer, and train our model in 3 phases with increasing sequence lengths. The input sequence lengths are 2048, 4096 and 8192 respectively for the 3 phases. By comparison, Longformer trains their model in 5 phases on GPUs with 48GB memory (The maximal of ours is 32GB) where the sequence length is 23,040 in the last phase. The window size of Longformer increases with depth and its average window size is 4352 in phase 5, while our effective number of attended tokens is 1280 on average in the last phase. Each experiment takes around 8 days to finish on 8 V100 GPUs. Detailed hyperparameters are shown in Appendix D. For testing, same as Longformer, we split the dataset into overlapping sequences of length 32K at a step size of 512, and evaluate the BPCs for predicting the next 512 tokens given the previous 32K characters. ",
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+        "type": "text",
+        "text": "Results Table 4 shows comparisons on text8 and enwik8. Our method has achieved state-of-the-art results. On text8, we achieve a test BPC of 1.09 with the smaller model. On enwik8, our smaller model achieves a test BPC of 0.99, and outperforms the state-of-the-art models with comparable number of parameters. Our larger model obtains a test BPC of 0.97, on par with the Compressive Transformer with $2 \\times$ parameters. Our results are consistently better than Longformer which is trained on longer sequences with 5 stages and 48 GPU memory. In Figure 3, we show our model is much more memory and computational efficient than full attention. ",
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+        "table_caption": [
+            "Table 5: Test accuracies on ImageNet, ImageNet Real [52], and ImageNet V2 [53] of models trained on ImageNet-1K. Grey-colored rows are our results. ${ \\mathrm { C v T ^ { * } } }$ -LS denotes our long-short term attention based on the non-official CvT implementation. ViL models with LS suffixes are our long-short term attention based on the official ViL implementation with relative positional bias. We also provide the latency of models tested using batch size 32 on the same V100 GPU. Our improvements over ViL is mainly from a better implementation of the short-term attention. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>(M)</td><td>Size</td><td>#Param Image FLOPs ImageNet (G)</td><td>top-1 (%)</td><td>Real top-1 (%) top-1 (%)</td><td>V2</td><td>Latency (s）</td></tr><tr><td>ResNet-50</td><td>25</td><td>224²</td><td>4.1</td><td>76.2</td><td>82.5</td><td>63.3</td><td>=</td></tr><tr><td>ResNet-101</td><td>45</td><td>2242</td><td>7.9</td><td>77.4</td><td>83.7</td><td>65.7</td><td></td></tr><tr><td>ResNet-152</td><td>60</td><td>2242</td><td>11</td><td>78.3</td><td>84.1</td><td>67.0</td><td>=</td></tr><tr><td>DeiT-S [36]</td><td>22</td><td>224²</td><td>4.6</td><td>79.8</td><td>85.7</td><td>68.5</td><td></td></tr><tr><td>DeiT-B [36]</td><td>86</td><td>2242</td><td>17.6</td><td>81.8</td><td>86.7</td><td>70.9</td><td></td></tr><tr><td>PVT-Medium [5]</td><td>44</td><td>2242</td><td>6.7</td><td>81.2</td><td>1</td><td>1</td><td></td></tr><tr><td>PVT-Large [5]</td><td>61</td><td>2242</td><td>9.8</td><td>81.7</td><td></td><td></td><td></td></tr><tr><td>Swin-S [38]</td><td>50</td><td>2242</td><td>8.7</td><td>83.2</td><td></td><td></td><td></td></tr><tr><td>Swin-B[38]</td><td>88</td><td>2242</td><td>15.4</td><td>83.5</td><td></td><td></td><td>0.115</td></tr><tr><td>PVTv2-B4 [54]</td><td>62.6</td><td>2242</td><td>10.1</td><td>83.6</td><td></td><td></td><td></td></tr><tr><td>PVTv2-B5 [54]</td><td>82.0</td><td>2242</td><td>11.8</td><td>83.8</td><td></td><td></td><td></td></tr><tr><td>ViT-B/16 [4]</td><td>86</td><td>3842</td><td>55.5</td><td>77.9</td><td></td><td></td><td></td></tr><tr><td>ViT-L/16 [4]</td><td>307</td><td>3842</td><td>191.1</td><td>76.5</td><td></td><td></td><td></td></tr><tr><td>DeiT-B [36]</td><td>86</td><td>3842</td><td>55.5</td><td>83.1</td><td></td><td></td><td></td></tr><tr><td>Swin-B [38]</td><td>88</td><td>3842</td><td>47.1</td><td>84.5</td><td>1</td><td>1</td><td>0.378</td></tr><tr><td>CvT-13 [6]</td><td>20</td><td>2242</td><td>6.7</td><td>81.6</td><td>86.7</td><td>70.4</td><td>0.122</td></tr><tr><td>CvT-21 [6]</td><td>32</td><td>2242</td><td>10.1</td><td>82.5</td><td>87.2</td><td>71.3</td><td>0.165</td></tr><tr><td>CvT*-LS-13</td><td>20.3</td><td>2242</td><td>4.9</td><td>81.9</td><td>87.0</td><td>70.5</td><td>0.083</td></tr><tr><td>CvT*-LS-17</td><td>23.7</td><td>2242</td><td>9.8</td><td>82.5</td><td>87.2</td><td>71.6</td><td>-</td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>224²</td><td>7.9</td><td>82.7</td><td>87.5</td><td>71.9</td><td>0.122</td></tr><tr><td>CvT*-LS-21S</td><td>30.1</td><td>2242</td><td>11.3</td><td>82.9</td><td>87.4</td><td>71.7</td><td>-</td></tr><tr><td>CvT-13 [6]</td><td>20</td><td>3842</td><td>31.9</td><td>83.0</td><td>87.9</td><td>71.9</td><td>1</td></tr><tr><td>CvT-21 [6]</td><td>32</td><td>3842</td><td>45.0</td><td>83.3</td><td>87.7</td><td>71.9</td><td></td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>3842</td><td>23.9</td><td>83.2</td><td>88.0</td><td>72.5</td><td></td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>4482</td><td>34.2</td><td>83.6</td><td>88.2</td><td>72.9</td><td></td></tr><tr><td>ViL-Small [14]</td><td>24.6</td><td>2242</td><td>4.9</td><td>82.4</td><td>-</td><td>1</td><td></td></tr><tr><td>ViL-Medium [14]</td><td>39.7</td><td>2242</td><td>8.7</td><td>83.5</td><td></td><td></td><td>0.106</td></tr><tr><td>ViL-Base [14]</td><td>55.7</td><td>2242</td><td>13.4</td><td>83.7</td><td></td><td></td><td>0.164</td></tr><tr><td>ViL-LS-Medium</td><td>39.8</td><td>2242</td><td>8.7</td><td>83.8</td><td></td><td></td><td>0.075</td></tr><tr><td>ViL-LS-Base</td><td>55.8</td><td>224²</td><td>13.4</td><td>84.1</td><td></td><td></td><td>0.113</td></tr><tr><td> ViL-LS-Medium</td><td>39.9</td><td>3842</td><td>28.7</td><td>84.4</td><td></td><td></td><td>0.271</td></tr></table>",
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+        "text": "4.3 ImageNet Classification ",
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+        "text": "We train and evaluate the models on ImageNet-1K with $1 . 3 \\mathbf { M }$ images and 1K classes. We use CvT [6] and ViL [11], state-of-the art vision transformer architectures, as the backbones and replace their attention mechanisms with our long-short term attention, denoted as ${ \\mathrm { C v T ^ { * } } }$ -LS and ViL-size-LS in Table 5. CvT uses overlapping convolutions to extract dense patch embeddings from the input images and feature maps, resulting in a long sequence length in the early stages (e.g., $5 6 \\times 5 6 = 3 1 3 6$ patches for images with $2 2 4 ^ { 2 }$ pixels). For ViL, our sliding window uses the same group size $w$ , but each token attends to at most $2 w \\times 2 w$ (rounding when necessary) tokens inside the window, instead of $3 w \\times 3 w$ as ViL, which allows adding our dynamic projection without increasing the FLOPs. We set $r = 8$ for the dynamic projections for both ViL-LS-Medium and ViL-LS-Base. Note that, our efficient attention mechanism does not depend on the particular architecture, and it can be applied to other vision transformers [e.g., 4, 36, 5]. Please refer to Appendix E for more details. ",
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+    {
+        "type": "text",
+        "text": "Classification Results. The results are shown in the Table 5, where we also list test accuracies on ImageNet Real and ImageNet V2. Except for CvT, we compare with the original ViT [4] and the enhanced DeiT [36], PVT [5] that also uses multi-scale stragey, ViL [11] that uses window attention and global tokens to improve the efficiency. Training at high-resolution usually improves the test accuracy of vision transformer. With our long-short term attention, we can easily scale the training to higher resolution, and the performance of ${ \\mathrm { C v T ^ { * } } }$ -LS and ViL-LS also improves. Our best model with CvT ${ \\mathrm { C v T ^ { * } } }$ -LS-21 at $4 4 { \\bar { 8 } } ^ { 2 }$ ) achieves $0 . 3 \\%$ higher accuracy than the best reported result of CvT while using the same amount of parameters and $76 \\%$ of its FLOPs. In CvT architecture, the spatial dimension of feature maps in earlier stages are large, representing more fine-grained details of the image. Similar to training with high-resolution images, the model should also benefit from denser feature maps. With our efficient long-short term attention, we can better utilize these fine-grained feature maps with less concerns about the computational budget. In this way, our ${ \\mathrm { C v T ^ { * } } }$ -LS-17 achieves better result than CvT-21 at resolution 224 using fewer parameters and FLOPs, and our ${ \\mathrm { C v T ^ { * } } }$ -LS-21S model further improves our ${ \\mathrm { C v T ^ { * } } }$ -LS-21 model. ",
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+    {
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+        "table_caption": [
+            "Table 6: Robustness evaluation on various ImageNet datasets. Top-1/Acc.: Top-1 accuracy. mCE: Mean Corrupution Error. Mixed-same/Mixed-rand: accuracies on MIXED-SAME/MIXED-RAND subsets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td colspan=\"5\">[Params|ImageNet|IN-C [56]|IN-A [57]|IN-R [58]]</td><td colspan=\"2\">ImageNet-9 [59]</td></tr><tr><td></td><td>(M)</td><td>Top-1</td><td>mCE(↓）</td><td>Acc.</td><td>Acc.</td><td>|Mixed-same Mixed-rand</td><td></td></tr><tr><td>ResNet-50 [35]]</td><td>25.6</td><td>76.2</td><td>78.9</td><td>6.2</td><td>35.3</td><td>87.1</td><td>81.6</td></tr><tr><td>DeiT-S [36]</td><td>22.1</td><td>79.8</td><td>57.1</td><td>19.0</td><td>41.9</td><td>89.1</td><td>84.2</td></tr><tr><td>CvT-13</td><td>20</td><td>81.6</td><td>59.6</td><td>25.4</td><td>42.9</td><td>90.5</td><td>85.7</td></tr><tr><td>CvT-21</td><td>32</td><td>82.5</td><td>56.2</td><td>31.1</td><td>42.6</td><td>90.5</td><td>85.0</td></tr><tr><td>CvT*-LS-13</td><td>20.3</td><td>81.9</td><td>58.7</td><td>27.0</td><td>42.6</td><td>90.7</td><td>85.6</td></tr><tr><td>CvT*-LS-21</td><td>32.1</td><td>82.7</td><td>55.2</td><td>29.3</td><td>45.0</td><td>91.5</td><td>85.8</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Our ViL-LS-Medium and ViL-LS-Base with long-short term attention improve the accuracies of ViL-Medium and ViL-Base from 83.5 and 83.7 to 83.8 and 84.1 respectively, without an increase in FLOPs. When increasing the resolution for training ViL-LS-Medium from $\\dot { 2 } 2 4 ^ { 2 }$ to $3 8 4 ^ { 2 }$ , the FLOPs increased (approximately) linearly and the accuracy improved by $0 . 6 \\%$ , showing our method still benefits greatly from increased resolution while maintaining the linear complexity in practice. ",
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+    {
+        "type": "text",
+        "text": "Short-term Attention Suppresses Oversmoothing. By restricting tokens from different segments to attend to different windows, our short-term sparse local attention encourages diversity of the feature representations and helps to alleviate the over-smoothing problem [55] (where all queries extract similar information in deeper layers and the attention mechanism is less important), thus can fully utilize the depth of the network. As in [55], we provide the cosine similarity of patch embeddings of our ${ \\mathrm { C v T ^ { * } } }$ -LS-13 and re-implemented CvT-13 (81.1 accuracy) in Figure 6 within Appendix. This is one of the reasons why our efficient attention mechanism can get even better results than the full attention CvT model in the same setting. ",
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+    {
+        "type": "text",
+        "text": "Robustness evaluation on Diverse ImageNet Datasets. As vision models have been widely used in safety-critical applications (e.g. autonomous driving), their robustness is vital. In addition to out-of-distribution robustness (ImageNet-Real and Imageet-v2), we further investigate the robustness of our vision transformer against common corruption (ImageNet-C), semantic shifts (ImageNet-R), Background dependence (ImageNet-9) and natural adversarial examples (ImageNet-A). We compare our methods with standard classification methods, including CNN-based model (ResNet [35]) and Transformer-based models (DeiT [36]) with similar numbers of parameters. As shown in Table 6, we observe that our method significantly outperforms the CNN-based method (ResNet-50). Compared to DeiT, our models also achieve favorable improvements. These results indicate that the design of different attention mechanisms plays an important role for model robustness, which sheds new light on the design of robust vision transformers. More details and results can be found in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we introduced Long-Short Transformer, an efficient transformer for long sequence modeling for both language and vision domain, including both bidirectional and autoregressive models. We design a novel global attention mechanism with linear computational and memory complexity in sequence length based on a dynamic projection. We identify the scale mismatch issue and propose the DualLN technique to eliminate the mismatch at initialization and more effectively aggregate the local and global attentions. We demonstrate that our method obtains the state-of-the-art results on the Long Range Arena, char-level language modeling and ImageNet classification. We look forward to extending our methods to more domains, including document QA, object detection and semantic segmentation on high-resolution images. ",
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+        "type": "text",
+        "text": "References   \n[1] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NIPS, volume 30, 2017.   \n[2] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. NAACL, 2019.   \n[3] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.   \n[4] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In ICLR, 2021.   \n[5] Wenhai Wang, Enze Xie, Xiang Li, Deng-Ping Fan, Kaitao Song, Ding Liang, Tong Lu, Ping Luo, and Ling Shao. 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In ACL, 2011.   \n[64] Dragomir R Radev, Pradeep Muthukrishnan, Vahed Qazvinian, and Amjad Abu-Jbara. The acl anthology network corpus. Language Resources and Evaluation, 47(4):919–944, 2013.   \n[65] Ilya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983, 2016.   \n[66] François Chollet. Xception: Deep learning with depthwise separable convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1251–1258, 2017.   \n[67] Lei Huang, Xianglong Liu, Yang Liu, Bo Lang, and Dacheng Tao. Centered weight normalization in accelerating training of deep neural networks. In Proceedings of the IEEE International Conference on Computer Vision, pages 2803–2811, 2017.   \n[68] Siyuan Qiao, Huiyu Wang, Chenxi Liu, Wei Shen, and Alan Yuille. Micro-batch training with batch-channel normalization and weight standardization. arXiv preprint arXiv:1903.10520, 2019. ",
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parse/train/P5MtdcVdFZ4/P5MtdcVdFZ4.md

@@ -0,0 +1,313 @@
+# AugMax: Adversarial Composition of Random Augmentations for Robust Training
+
+Haotao Wang1∗, Chaowei $\mathbf { X i a o ^ { 2 , 3 } }$ , Jean Kossaifi2, Zhiding $\mathbf { Y } \mathbf { u } ^ { 2 }$ , Anima Anandkumar2,4, and Zhangyang Wang1
+
+1Department of Electrical and Computer Engineering, University of Texas at Austin 2NVIDIA 3Arizona State University 4California Institute of Technology 1{htwang, atlaswang}@utexas.edu 2{chaoweix, jkossaifi, zhidingy, aanandkumar}@nvidia.com
+
+# Abstract
+
+Data augmentation is a simple yet effective way to improve the robustness of deep neural networks (DNNs). Diversity and hardness are two complementary dimensions of data augmentation to achieve robustness. For example, AugMix explores random compositions of a diverse set of augmentations to enhance broader coverage, while adversarial training generates adversarially hard samples to spot the weakness. Motivated by this, we propose a data augmentation framework, termed AugMax, to unify the two aspects of diversity and hardness. AugMax first randomly samples multiple augmentation operators and then learns an adversarial mixture of the selected operators. Being a stronger form of data augmentation, AugMax leads to a significantly augmented input distribution which makes model training more challenging. To solve this problem, we further design a disentangled normalization module, termed DuBIN (Dual-Batch-and-Instance Normalization), that disentangles the instance-wise feature heterogeneity arising from AugMax. Experiments show that AugMax-DuBIN leads to significantly improved out-of-distribution robustness, outperforming prior arts by $3 . 0 3 \%$ , $3 . 4 9 \%$ , $1 . 8 2 \%$ and $0 . 7 1 \%$ on CIFAR10-C, CIFAR100-C, Tiny ImageNet-C and ImageNet-C. Codes and pretrained models are available: https://github.com/VITA-Group/AugMax.
+
+# 1 Introduction
+
+Out-of-distribution (OOD) samples present a challenge when deploying AI models in the real world. Examples include natural corruptions (e.g., due to camera blurs or noise, snow, rain, or fog in image data), sensory perturbations (e.g., sensor transient error, electromagnetic interference) and domain shifts (e.g., summer winter). However, deep networks are often trained on limited amounts of data which may not cover sufficient scenarios. As a result, they are vulnerable to unforeseen distributional changes despite achieving high performance on standard benchmarks [1–3]. This jeopardizes their trustworthiness as well as safe deployment in real-world environments. Thus it is critical to develop techniques that improve robustness even when training with relatively clean datasets.
+
+Several techniques have been proposed to consolidate the robustness against unforeseen corruptions, including robust data augmentation [5, 3, 8, 6], Lipschitz continuity [9–11], stability training [12], pre-training [13–16], and robust network structures [17–19], to name a few. Among these techniques, data augmentation is of particular interest due to its empirical effectiveness, ease of implementation, low computational overhead, and plug-and-play nature.
+
+![](images/96da970a0935f651f016706faf1673e344b89a5c13bdb2ecde02923b0a6543ea.jpg)  
+Figure 1: The effects of diversity and hardness in data augmentations. We visualize features of augmented images fed to the network during training. The features are from the penultimate layer of a ResNeXt29 trained on CIFAR10 training data using only standard data augmentation (random flipping and translation) following [4]. For visualization, we randomly selected 300 fixed images from 3 fixed classes (denoted by different colors) to which we apply different augmentation methods: (a) standard augmentation (random flipping and translation); (b) AugMix [5]; (c) PGD Attack [6, 7]; (d) AugMax (ours). In order to achieve good model robustness, the augmented training data should both be diverse and also contain enough hard cases. As can be seen, PGD Attack generates hard cases (which the network cannot separate) but are not diverse enough (they are all clustered together) while AugMix creates diverse but not hard samples (they are well separated). By contrast, our approach (AugMax) achieves a unification between hard and diverse samples.
+
+There are mainly two categories of data augmentation approaches:
+
+The first category aims to increase diversity of the training data by composing multiple random transformations [20–22, 5, 3]. While standard data augmentation methods (e.g., random flipping and translation) lead to poor robustness [5], more aggressive combinations of multiple augmentations have shown promise. One such successful example is AugMix [5], which stochastically samples from diverse augmentation operations and randomly mixing them to produce highly diverse augmented images. AugMix can increase the sample diversity coverage compared to standard augmentations, as shown in Figure 1 (a) and (b).
+
+The second category aims to boost the hardness of the training data by sampling from the worst-case augmentations that tries to fool the model into misclassifying the samples. A common technique to achieve this goal is adversarial perturbation [6, 23, 8]. Training over such worst-case samples allows a model to actively fix its generalization weaknesses [24] and empirically improves its robustness against corruptions [25, 26, 7]. Due to the extra complexity to generate adversarial perturbations, the improved robustness usually comes at the cost of largely increased training time compared with non-adversarial methods [6, 27]. An example falling into this category is the PGD attack [6], as is shown in Figure 1 (c).
+
+Previous work has focused on leveraging one of these category to improve robustness. In this paper, we unify both approaches in a single framework. In particular, we show that diversity and hardness are complementary and that a unification between the two is necessary to achieve robustness. We propose a new strategy to achieve this unification and successfully increase robustness.
+
+# Summary of contributions:
+
+• We propose AugMax, a novel augmentation framework which achieves robustness through a unification between diversity and hardness, by searching for the worst-case mixing strategy. • Being a stronger form of data augmentation, AugMax leads to a significantly augmented and more heterogeneous input distribution, which also makes model training more challenging. To solve this problem, we design a new normalization strategy, termed DuBIN, to disentangle the instance-wise feature heterogeneity of AugMax samples. • We show that combination of AugMax and DuBIN (AugMax-DuBIN) achieves state-of-the-art robustness against corruptions and improves robustness against other common distribution shifts.
+
+AugMax achieves a good unification between sample diversity and hard corner-case generation during data augmentation. AugMax is built on top of the AugMix framework [5] which mixes multiple data augmentation operators in a multi-branch and layered pipeline. However, different from AugMix where augmentation operators and mixing weights are both randomly sampled, operators are first randomly sampled, followed by adversarially trained mixture of the selected operators in Augmax. This simple change from AugMix to AugMax results in considerable difference in their feature distributions. From the visualizations (in Figure 1), AugMax generates more adversarially “hard samples", while still keeping a good amount of diversity compared to AugMix and PGD Attack. Searching for adversarial mixing strategies in AugMax is slightly more expensive since it involves adversarial mixing. To make this efficient, we adopt an efficient adversarial training strategy [28]. As a result, AugMax adds only a reasonable amount of extra complexity compared to non-adversarial training methods, while improving robustness significantly. For example, AugMax training time is only $\sim 1 . 5$ times that of AugMix on ImageNet (see Table 7). Moreover, Augmax is significantly more efficient than traditional adversarial training which has $\sim 1 0$ times training time overhead compared with AugMix.
+
+![](images/5c2b2929baf2ee2ecdb10d564875ad242f4ebe5bd3f8d0fcce5c19740424a479.jpg)  
+Figure 2: AugMax overview. Black and red arrows represent forward paths and back-propagation paths to generate AugMax images, respectively. In contrast to AugMax, where the mixing parameters $m$ and $\pmb { w }$ are adversarially learned, AugMix randomly samples $m$ and $\pmb { w }$ from predefined distributions (and thus no backpropergation on $m$ and $\textbf { \em w }$ ).
+
+Being a stronger form of data augmentation with adversarial sample generation, AugMax leads to a significantly augmented input distribution which makes model training more challenging. This naturally motivates us to propose a novel and finer-grained normalization scheme termed Dual-Batchand-Instance Normalization (DuBIN). As illustrated in Figure 3, DuBIN adds an additional instance normalization (IN) in parallel to the traditional Dual Batch Normalization (DuBN) [7, 29] used in adversarial training, in order to better model and disentangle the instance-wise feature heterogeneity arising from AugMax. We show that adding the instance normalization is an important knob to promote the capability of AugMax in boosting model robustness.
+
+Our framework, AugMax-DuBIN, is illustrated in Figure 2. AugMax-DuBIN trains on clean images and achieves state-of-the-art robustness on natural corruption benchmarks [1], and also improves model robustness against other common distribution shifts [30, 2]. In particular, our method surpasses state-of-the-art method on CIFAR10-C, CIFAR100-C, Tiny ImageNet-C and ImageNet-C by $3 . 0 3 \%$ , $3 . 4 9 \%$ , $1 . 8 2 \%$ and $0 . 7 1 \%$ respectively.
+
+# 2 Related Work
+
+# 2.1 Robustness to Distributional Shifts
+
+Hendrycks et al. [1] pioneer the study of prediction errors exhibited by machine learning models on unseen natural corruptions and perturbations. They introduce two variants of the original ImageNet validation set to evaluate the model’s robustness: the ImageNet-C for input corruption robustness, and the ImageNet-P for input perturbation robustness. The former applies 15 diverse corruptions drawn from four main categories (noise, blur, weather, and digital) on ImageNet validation images. The latter contains perturbation sequences generated from each image. Recht et al. [30, 2] collect CIFAR10.1 and ImageNet-V2 as a reproduction of the original CIFAR10 [31] and ImageNet [32] test sets, and find that a minute natural distribution shift caused by the minutiae in dataset collection process leads to a large drop in accuracy for a broad range of image classifiers. Geirhos et al. [33] observe that deep models trained on ImageNet are biased towards textures, and the model robustness can be improved by emphasizing more on global shape features [33–36]. Wang et al. [37] find that the benchmark test performance does not always translate to the real-world generalizability, and propose an efficient method to troubleshoot trained models using real-world unlabeled images. Hendrycks et al. [38] collect ImageNet-A with the new concept of natural adversarial examples: unmodified real-world images that falsify common machine learning models. Other representative benchmark efforts include [39–42, 3], all demonstrating the brittleness of deep models under various distribution shifts. Readers of interest are referred to a recent survey [43].
+
+# 2.2 Data Augmentation: Random and Adversarial
+
+Data augmentations have been widely used to increase training set diversity and improve model generalization ability [44, 21, 33, 22]. Among those methods, AugMix [5] achieves the outstanding robustness against natural corruptions by randomly mixing multiple diverse augmentations. It establishes high performance bars on many natural corruption benchmarks including CIFAR10-C, CIFAR100-C and ImageNet-C [1]. DeepAugment [3] feeds clean images to image-to-image models with randomly perturbed model weights to generate visually diverse augmented images. Combined with AugMix, it achieves state-of-the-art robustness on ImageNet-C [1]. Lately, a concurrent work MaxUp [45] generates hard training samples by selecting the worst-case weights in the MixUp [20] pipeline, to improve both model generalization ability and adversarial robustness. Our work shared a similar mindset to MaxUp, but we build on the more sophisticated AugMix pipeline and handle the resulting higher feature heterogeneity using a new normalization module.
+
+Besides, Adversarial training (AT) [6, 46–52] utilizes adversarial samples as a special data augmentation method, and has also shown promise in improving model robustness against natural corruptions [23] and domain gaps [25], usually at the cost of increasing training time [27] and degrading the standard accuracy on clean images [53]. Volpi et al. [25] and Zhao et al. [26] augment the training set with samples from a fictitious adversarial domain to improve domain adaptation performance. Xie et al. [7] show that adversarial training can improve both standard accuracy and robustness against distribution shifts, with the help of auxiliary batch normalization. Adversarial Noise Training (ANT) [8] uses random Gaussian noise with learned adversarial hyperparameters (mean and variance) as the augmentation. Gowal et al. [54] propose to generate new augmented images by adversarially composing the representations of different original images. Wang et al. [55] improve the robustness of pose estimation models by adversarially mixing different corrupted images as data augmentation. Based on DeepAugment [3], Calian et al. [56] propose AdversarialAugment, which optimizes the parameters of image-to-image models to generate adversarially augmented images, achieving promising results against natural corruptions. Robey et al. [57] proposed model-based robust learning which generates augmented images with adversarial natural variations to improve model robustness against naturally-occurring conditions (e.g., snow, decolorization, shadow, and many others). The authors further generalized such idea to the domain generalization problem and achieved impressive results [58].
+
+# 2.3 Normalization
+
+Batch Normalization (BN) [59] has been a cornerstone for training deep networks. Inspired by BN, more task-specific modifications are proposed by exploiting different normalization axes, such as Instance Normalization (IN) [60], Layer Normalization (LN) [61] and Group Normalization (GN) [62]. A few latest works investigate to use multiple normalization layers instead of one BN. Zajkac et al. [63] use two separate BNs to handle the domain shift between labeled and unlabeled data in semi-supervised learning. Xie et al. [64, 7] observe the difference between standard and adversarial feature statistics during AT, and craft dual batch normalization (DuBN) to disentangle the standard and adversarial feature statistic to improve both the standard accuracy and robustness. While most deep models just use one normalization type throughout the network, IBN-Net [29] unifies IN and BN in one model. It finds that when applied together, IN tends to learn features invariant to appearance (colors, styles, etc.) changes, while BN is essential for preserving high-level semantic contents. IBN-Net shows that combining IN and BN in an appropriate manner can improve model generalization. A similar combination was also explored in the style transfer field [65], learning to selectively normalize only disturbing styles while preserving useful styles.
+
+# 3 Method
+
+# 3.1 Preliminary: The AugMix Framework
+
+At a high level, AugMix creates a new composite image from a clean sample by first applying several simple augmentation operations to it before combining the results through random linear combinations. This augmentation scheme is coupled with a Jensen-Shannon consistency loss that enforces feature similarity among clean and augmented images. Specifically, AugMix consists of multiple augmentation chains (3 by default), each applied in parallel to the same input image. Each chain composes one to three randomly selected augmentation operations. These augmentations are the same as AutoAugment [21], but excluding those that overlap with ImageNet-C corruptions. The augmented images from each augmentation chains are then combined together with the original, clean sample using a linear combinations with randomly samples coefficients. The final image thus incorporates several sources of randomness, coming respectively from the choice of operations, the severity level of these operations, the lengths of the augmentation chains and the mixing weights. Despite its simplicity, AugMix achieves state-of-the-art corruption robustness. In the remaining of this section, we build on the previous AugMix work and propose AugMax, an augmentation strategy that further improves robustness.
+
+# 3.2 AugMax: Augmented Training with Unified Diversity and Hardness
+
+In this section, we rigorously introduce our new data augmentation method, AugMax, illustrated in Figure 2. At a high level, the main difference between AugMax and AugMix is a new optimization procedure used to select the mixing weights $m$ and $\pmb { w }$ .
+
+Given a data distribution $\mathbb { D }$ over images $\pmb { x } \in \mathbb { R } ^ { d }$ and labels $\pmb { y } \in \mathbb { R } ^ { c }$ , our goal is to train a mapping (classifier) $f : \mathbb { R } ^ { d }  \mathbb { R } ^ { c }$ from images to output softmax probabilities, parameterized by $\pmb \theta$ , that is robust to (unknown) distribution shifts. We minimize the empirical risk
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim \mathcal { D } } \ \mathcal { L } ( f ( \pmb { x } ; \pmb { \theta } ) , \pmb { y } ) ,
+$$
+
+where $\mathcal L ( \cdot , \cdot )$ is the loss function (e.g., cross-entropy). As illustrated in Figure 2, an AugMax image $\pmb { x } ^ { * }$ is generated by learning a set of adversarial mixing parameters $m ^ { * } , w ^ { * }$ which maximizes the loss:
+
+$$
+{ \pmb x } ^ { * } = g ( { \pmb x } _ { o r i g } ; m ^ { * } , { \pmb w } ^ { * } ) ,
+$$
+
+where $g ( \cdot )$ denotes the AugMax augmentation function, $\pmb { x } _ { o r i g }$ is the original image, and
+
+$$
+m ^ { * } , \boldsymbol { w } ^ { * } = \underset { m , w } { \arg \operatorname* { m a x } } \mathcal { L } \big ( f ( g ( x _ { o r i g } ; m , w ) ; \theta ) , y \big ) , \quad \mathrm { s . t . } m \in [ 0 , 1 ] , w \in [ 0 , 1 ] ^ { b } , w ^ { T } \mathbf { 1 } = 1 .
+$$
+
+To simplify the optimization problem in Eq. (3), we use a re-parameterization trick by setting ${ \pmb w } = \sigma ( { \pmb p } )$ , where $\sigma ( \cdot )$ is the softmax function, and convert the optimization into a differentiable one:
+
+$$
+m ^ { * } , \pmb { p } ^ { * } = \operatorname * { a r g m a x } _ { m \in [ 0 , 1 ] , \pmb { p } \in \mathbb { R } ^ { b } } \mathcal { L } ( f ( g ( \pmb { x } _ { o r i g } ; m , \sigma ( \pmb { p } ) ) ; \theta ) , \pmb { y } ) ; \quad \pmb { w } ^ { * } = \sigma ( \pmb { p } ^ { * } )
+$$
+
+Training with AugMax augmentation can then be written altogether as minimax optimization:
+
+$$
+\begin{array} { r l } & { \underset { \theta } { \operatorname* { m i n } } \mathbb { E } _ { ( \boldsymbol { x } , \boldsymbol { y } ) \sim \mathcal { D } } \frac { 1 } { 2 } [ \mathcal { L } ( f ( \boldsymbol { x } ^ { * } ) ; \boldsymbol { \theta } ) , \boldsymbol { y } ) + \mathcal { L } ( f ( \boldsymbol { x } ) ; \boldsymbol { \theta } ) , \boldsymbol { y } ) ] + \lambda \mathcal { L } _ { c } ( \boldsymbol { x } , \boldsymbol { x } ^ { * } ) } \\ & { \mathrm { s . t . } \boldsymbol { x } ^ { * } = g ( \boldsymbol { x } ; m ^ { * } , \boldsymbol { w } ^ { * } ) ; \boldsymbol { w } ^ { * } = \sigma ( p ^ { * } ) ; \boldsymbol { m } ^ { * } , p ^ { * } = \underset { m \in [ 0 , 1 ] , p \in \mathbb { R } ^ { b } } { \arg \operatorname* { m a x } } \mathcal { L } ( f ( g ( \boldsymbol { x } ; m , \sigma ( p ) ) ; \boldsymbol { \theta } ) , \boldsymbol { y } ) } \end{array}
+$$
+
+where $\mathcal { L } _ { c }$ is a consistency loss regularizing augmented images to have similar model outputs with the original images and $\lambda$ is the trade-off parameter. Our implementation of $\mathcal { L } _ { c }$ is adapted from [5]:
+
+$$
+\mathcal { L } _ { c } ( { \pmb x } , { \pmb x } ^ { * } ) = \mathrm { J S } ( f ( { \pmb x } ; \theta ) , f ( \tilde { { \pmb x } } ; \theta ) , f ( { \pmb x } ^ { * } ; \theta ) )
+$$
+
+with $\mathrm { J } \mathrm { S } ( \cdot )$ being the Jensen-Shannon divergence and $\tilde { \pmb x }$ being the augmented image generated by AugMix from $_ { \textbf { \em x } }$ . In order to reduce the training complexity overhead caused by adversarial training, we employ an accelerated adversarial attack method [28] to solve Eq. (4), which adds only a reasonable amount of extra complexity compared to AugMix baseline. The algorithm to solve Eq. (5) is summarized in Appendix A.
+
+Visualization of the effects of diversity and hardness To illustrate the motivation behind AugMax, we visualize the feature representations induced by different augmentation methods to showcase their diversity and hardness in Figure 1. Specifically, we visualize the impact of various data augmentation methods on the feature representations obtained with a ResNeXt29 that is normally trained on CIFAR-10. Note that the model is trained using only default standard augmentations (i.e., random flipping and translation), and therefore has never seen augmentation from AugMix, AugMax, or PGD attack. Following [4], we visualize the penultimate layer feature distributions of training samples from three different classes. Three hundred training images are selected from each class for visualization. For AugMix and $\ell _ { \infty }$ PGD attack, we use the same hyperparameters in the original papers [5, 6].
+
+We observe in Figure 1 that the clean images form well-separated clusters in the feature space, leaving large regions between those clusters empty. This potentially causes uncertain and unreliable generalization if new test samples (from a shifted distribution) fall into those regions. AugMix enlarges each cluster to cover a broader region, effectively increasing sample diversity. However, after augmentation, very few samples are close to the decision boundaries, revealing an inability to generate hard samples. Samples from the PGD attack, on the other hand, collapse into a small region around the classification borders, losing feature diversity. We further investigate the method through ablation studies in Section 4.3.
+
+# 3.3 DuBIN: Disentangled Normalization for Heterogeneous Features
+
+AugMax leads to better coverage of the input space by unifying sample diversity and hardness, as demonstrated in Figure 1. While such more versatile distribution in data augmentation has the potential to increase robustness, we do observe that naive incorporation of AugMax leads to relatively marginal improvement over AugMix (see Section 4.3). This is due to the comprehensiveness of AugMax can also lead to a higher feature heterogeneity, which may require larger model capacity to encode. To address this problem, we design a novel normalization layer, termed Dual Batch-and-Instance Normalization (DuBIN), to disentangle the instance-level heterogeneity.
+
+As illustrated in Figure 3, DuBIN consists of two parallel parts: (1) a dual batch normalization (DuBN) [7], which is the by-default normalization layer to use for adversarial training [7, 64], to disentangle the group-level statistics of the clean and corner-case augmented samples2; and (2) an additional instance normalization, to account for instancelevel feature statistics due to augmentation diversity. At each layer, the input feature is split into halves along the channel dimension: one half is then fed into IN, and the other fed into DuBN. Finally, the outputs of IN and DuBN are concatenated back along the channel dimension for the next layer.
+
+![](images/37442dadfb77175a367b93c847e84dc9a4c01bb733ad3830171f67604343a360.jpg)  
+Figure 3: Illustration of DuBIN. “Split” and “cat” represent splitting and concatenating on features along the channel dimension. The input feature $x$ with $C$ channels is split into halves (i.e., $x _ { A }$ and $x _ { B }$ , each with $C / 2$ channels). Each half goes into different normalization layers (either IN or DuBN) and the outputs are concatenated back along the channel dimension.
+
+To provide additional insights of DuBIN, we compare the BN statistics of AugMax-DuBN and AugMax-DuBIN in Table 1. $\overline { { \sigma } } _ { c } ^ { 2 }$ and $\overline { { \sigma } } _ { a } ^ { 2 }$ represent the average batch normalization variance over all channels of $B N _ { c }$ and $B N _ { a }$ , respectively. We observe that both $\overline { { \sigma } } _ { c } ^ { 2 }$ and $\overline { { \sigma } } _ { a } ^ { 2 }$ are smaller in the DuBIN network than those in the DuBN counterpart. This shows that the IN branch in DuBIN can reduce the feature diversity that BN otherwise needs to model, by
+
+Table 1: BN statistics of different layers in WRN40-2 with DuBN or DuBIN trained on CIFAR100 using AugMax training.   
+
+<table><tr><td colspan="2">Layer</td><td>Block 1 layer 2</td><td>Block1 layer 3</td><td>Block 2 layer 2</td></tr><tr><td rowspan="3">AugMax-DuBN</td><td></td><td>0.0369</td><td>0.0450</td><td>0.0301</td></tr><tr><td>品</td><td>0.0469</td><td>0.0585</td><td>0.0382</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">AugMax-DuBIN</td><td></td><td>0.0306</td><td>0.0403</td><td>0.0264</td></tr><tr><td></td><td>0.0348</td><td>0.0466</td><td>0.0292</td></tr><tr><td>R</td><td></td><td></td><td></td></tr></table>
+
+encoding instance-level diversities. With lower feature variation, BNs can converge better with improved performance [66].
+
+# 4 Experiments
+
+Here, we introduce in detail the experimental setting, and report quantitative results for robustness against natural corruptions and distribution shifts. We show that our method induces better robustness than existing ones. We study the properties of our approach in thorough ablation studies.
+
+# 4.1 Experimental Setup
+
+Datasets and Models We evaluate our proposed method on CIFAR10, CIFAR100 [31], ImageNet [32] and Tiny ImageNet $\left( \mathrm { T I N } \right)$ . For neural architectures, we use a ResNet18 [67], WRN40-2 [68] and ResNeXt29 [69] on the CIFAR datasets, and ResNet18 on ImageNet and Tiny ImageNet4. To evaluate the model’s robustness against common natural corruptions, we use CIFAR10-C, CIFAR100- C, ImageNet-C and Tiny ImageNet-C (TIN-C) [1], which are generated by adding natural corruptions to the original test set images.
+
+Evaluation Metrics Following established conventions [1, 5], we define robustness accuracy (RA) as the average classification accuracy over all 15 corruptions, to measure the models’ robustness on CIFAR10-C and CIFAR100-C. On ImageNet-C and Tiny ImageNet-C, we use both RA and mean corruption error (mCE) to evaluate robustness. As defined by [1], mCE is the weighted average of target model corruption
+
+Table 2: Evaluation results on CIFAR10 and CIFAR10-C. The best metric is shown in bold.   
+
+<table><tr><td>Model</td><td>Metric</td><td>Normal</td><td>AugMix</td><td>AugMax-DuBIN</td></tr><tr><td rowspan="2">ResNet18</td><td>SA(%)</td><td>95.56</td><td>95.79</td><td>95.76 (-0.03)</td></tr><tr><td>RA(%)</td><td>74.75</td><td>89.49</td><td>90.36 (+0.87)</td></tr><tr><td rowspan="2">WRN40-2</td><td>SA(%)</td><td>94.78</td><td>95.67</td><td>95.68 (+0.01)</td></tr><tr><td>RA(%)</td><td>73.71</td><td>89.01</td><td>90.67 (+1.66)</td></tr><tr><td rowspan="2">ResNeXt29</td><td>SA(%)</td><td>95.60</td><td>96.25</td><td>96.39 (+0.14)</td></tr><tr><td>RA(%)</td><td>71.70</td><td>89.08</td><td>92.11 (+3.03)</td></tr></table>
+
+errors normalized by the corruption errors of a baseline model across different types of corruptions. We use the AlexNet provide in [1] as the baseline model for experiments on ImageNet. Since [1] does not provide an off-the-shelf baseline model for Tiny ImageNet, we use the conventionally trained ResNet18 on Tiny ImageNet as the baseline model for experiments on Tiny ImageNet. We use standard accuracy (SA) to denote the classification accuracy on the original clean testing images.
+
+Baseline Methods On CIFAR10-C and CIFAR100-C, we compare our method with the state-of-the-art method, AugMix [5]. In addition, AugMix has previously been combined with other orthogonal methods, such as DeepAugment [3], to further boost performance on ImageNet-C. Specifically, [3] combine DeepAugment as an orthogonal component to AugMix, and achieve state-of-the-art
+
+Table 3: Evaluation results on CIFAR100 and CIFAR100-C. The best metric is shown in bold.   
+
+<table><tr><td>Model</td><td>Metric</td><td>Normal</td><td>AugMix</td><td>AugMax-DuBIN</td></tr><tr><td rowspan="2">ResNet18</td><td>SA(%)</td><td>77.99</td><td>78.23</td><td>78.69 (+0.46)</td></tr><tr><td>RA(%)</td><td>48.46</td><td>62.67</td><td>65.75 (+3.08)</td></tr><tr><td rowspan="2">WRN40-2</td><td>SA(%)</td><td>76.19</td><td>77.03</td><td>76.80 (-0.23)</td></tr><tr><td>RA(%)</td><td>46.80</td><td>64.56</td><td>66.35 (+1.79)</td></tr><tr><td rowspan="2">ResNeXt29</td><td>SA(%)</td><td>79.95</td><td>78.58</td><td>80.70 (+2.12)</td></tr><tr><td>RA(%)</td><td>47.76</td><td>65.37</td><td>68.86 (+3.49)</td></tr></table>
+
+performance on ImageNet-C with DeepAugment $^ +$ AugMix [3]. To allow comparison with this scenario, we also run the following comparison experiment on ImageNet-C and Tiny ImageNet-C: (1) AugMix v.s. AugMax and (2) DeepAugment $^ +$ AugMix v.s. DeepAugment $^ +$ AugMax. We show that the advantages of AugMax still exist when combined with other orthogonal methods.
+
+Finally, we also compare AugMax with ANT [8], which utilizes Gaussian noises with learned distribution parameters for data augmentation. Since models tend to generalize better on test images similar to those seen during training [70], ANT mainly improves robustness against high frequency corruptions (e.g., additive noises), while achieving comparable robustness on other unseen corruptions (e.g., bluring) with AugMix. Following [8], we perform the comparison on the subset of Tiny ImageNet-C with 12 remaining corruptions after removing all additive noises.
+
+Table 4: Evaluation results on ImageNet and ImageNet-C. The best metric is shown in bold.   
+
+<table><tr><td>Method</td><td>SA (%,↑)</td><td>RA (%,↑)</td><td>mCE(%,↓)</td></tr><tr><td>Normal</td><td>69.83</td><td>30.91</td><td>87.47</td></tr><tr><td>AugMix</td><td>68.06 67.62</td><td>34.58 35.01</td><td>83.08 82.56</td></tr><tr><td>AugMax-DuBIN</td><td>(-0.44)</td><td>(+0.43)</td><td>(-0.52)</td></tr><tr><td>DeepAugment + AugMix</td><td>65.32</td><td>45.84</td><td>69.29</td></tr><tr><td>DeepAugment +</td><td>64.43</td><td>46.55</td><td>68.47</td></tr><tr><td>AugMax-DuBIN</td><td>(-0.89)</td><td>(+0.71)</td><td>(-0.82)</td></tr></table>
+
+Implementation Details We follow [5] for hyperparameter settings. Specifically, for all experiments on CIAFR10 and CIFAR100, we use SGD optimizer with initial learning rate 0.1 and cosine annealing learning rate scheduler, and train all models for 200 epochs. For all experiments on ImageNet, we use SGD optimizer with initial learning rate 0.1 and batch size 256 to train the model for 90 epochs. We reduce the learning rate by $1 / 1 0$ at the 30-th and 60-th epoch. For all experiments on Tiny ImageNet, we train for 200 epochs and decay the learning rate at the 100-th and 150-th epoch. Other hyperparameter settings are identical as those in ImageNet experiments.
+
+We set batch size to 256 for all experiments. In Eq. (4), $m$ and $\pmb { p }$ are uniformly randomly initialized between 0 and 1. We use PGD to update $m$ : first do gradient ascend on $m$ and then project it back into the $[ 0 , 1 ]$ interval. $\pmb { p }$ is updated using gradient ascend. AugMax follows the same rules as AugMix [5] to randomly select augmentation operation type, severity level and chain length. Following [6], we iteratively solve the inner maximization and outer minimization problems in Eq. (5): the inner maximization is updated for $n$ steps, for every 1 step update on
+
+Table 5: Evaluation results on TIN and TIN-C. The best metric is shown in bold.   
+
+<table><tr><td>Method</td><td>SA (%,↑)</td><td>RA (%,↑)</td><td>mCE(%,↓)</td></tr><tr><td>Normal</td><td>61.64</td><td>23.91</td><td>100.00</td></tr><tr><td>AugMix</td><td>61.79 62.21</td><td>36.85 38.67</td><td>83.04 80.72</td></tr><tr><td>AugMax-DuBIN</td><td>(+0.42)</td><td>(+1.82)</td><td>(-2.32)</td></tr><tr><td>DeepAugment + AugMix</td><td>59.59</td><td>40.67</td><td>78.28</td></tr><tr><td>DeepAugment +</td><td>59.72</td><td>40.99</td><td>77.83</td></tr><tr><td>AugMax-DuBIN</td><td>(+0.13)</td><td>(+0.32)</td><td>(-0.45)</td></tr></table>
+
+the outer minimization. Both $m$ and $\pmb { p }$ are updated with step size $\alpha = 0 . 1$ . We set $n = 5$ on ImageNet for efficiency and $n = 1 0$ on other datasets. We set $\lambda$ in Eq. (5) to be 12 on ImageNet and 10 on all other datasets, except for ResNet18 on CIFAR100 where we find $\lambda = 1$ leads to better performance. All experiments are conducted on a server with four NVIDIA RTX A6000 GPUs.
+
+# 4.2 Robustness against Natural Corruptions
+
+Results on CIFAR10-C and CIFAR100-C are shown in Table 2 and 3 respectively, where “normal” means training with default standard augmentations (e.g., random flipping and translation). From the results, we see that AugMax-DuBIN achieves new state-of-the-art performance on both datasets with different model structures. For example, compared with AugMix, our method improves accuracy by as high as $3 . 0 3 \%$ and $3 . 4 9 \%$ on CIFAR10-C and CIFAR100-C, respectively. Moreover, we observe that our method benefits larger models more. Specifically, in both CIFAR10 and CIFAR100 experiments, the largest robustness gain is achieved on ResNeXt29, which has the largest capacity among the three models.
+
+Results on ImageNet are shown in Table 4. AugMax-DuBIN outperforms AugMix by $0 . 5 2 \%$ in terms of $\mathrm { m C E }$ on ImageNet-C. Further combining AugMax-DuBIN with DeepAugment outperforms AugMix $^ +$ DeepAugment by $0 . 8 2 \%$ in terms of mCE, achieving a new stateof-the-art performance on ImageNet-C. This
+
+Table 6: Evaluation results on TIN and TIN-C (w/o noise). The best metric is shown in bold.   
+
+<table><tr><td>Method</td><td>SA(%,↑)</td><td>RA(%,↑)</td><td>mCE(%,）</td></tr><tr><td>Normal</td><td>61.64</td><td>24.38</td><td>100.00</td></tr><tr><td>AugMix</td><td>61.79</td><td>37.63</td><td>82.54</td></tr><tr><td>ANT</td><td>61.26</td><td>35.30</td><td>85.70</td></tr><tr><td>AugMax-DuBIN</td><td>62.21</td><td>38.66</td><td>80.29</td></tr></table>
+
+shows that AugMax can be used as a more advanced basic building block for model robustness and inspires future researches to build other defense methods on top of it.
+
+Results on Tiny ImageNet-C are shown in Table 5 and Table 6. From Table 5, we see that our method improves the mCE by $2 . 3 2 \%$ compared with AugMix and $0 . 4 5 \%$ when combined with DeepAugment. From Table 6, we can see our method outperforms ANT by a considerable margin.
+
+Training Time Since we use an accelerated adversarial training method [28] to solve Eq. (4), the worst-case mixing strategy can be searched efficiently. As a result, AugMax adds only a modest training overhead over AugMix, and is significantly more efficient than traditional adversarial training (AT). Detailed training time is shown in Table 7.
+
+Table 7: Training time on ImageNet with ResNet18, reported on a single NVIDIA A6000.   
+
+<table><tr><td>Method</td><td>Time (sec/epoch)</td></tr><tr><td>Normal</td><td>2669</td></tr><tr><td>AugMix</td><td>3622</td></tr><tr><td>AT[6]</td><td>37162</td></tr><tr><td>AugMax</td><td>5264</td></tr></table>
+
+# 4.3 Ablation Study
+
+Analysis of Different Normalization Layers In this paragraph, we compare AugMax/AugMix when combined with different normalization schemes. When using DuBN or DuBIN, we route original images to the $B N _ { c }$ , and AugMax/AugMix images to $B N _ { a }$ . Since it is the by-default setting to use disentangled normalization layers (e.g., DuBN) for original and augmentation images in adversarial training [7, 64], we follow this traditional routing and combine AugMax with DuBN or DuBIN instead of BN or IBN. It is fair to compare AugMax-DuBN/DuBIN with either AugMixBN/IBN or AugMix-DuBN/DuBIN.
+
+Results are shown in Table 8. AugMax-DuBN outperforms both AugMix-BN and AugMixDuBN, and AugMax-DuBIN outperforms both AugMix-IBN and AugMix-DuBIN. Noticeably, when combined with AugMix, DuBIN also helps improve robustness. This shows the potential of applying DuBIN to other diversity augmentation methods for general performance improvement.
+
+We also conduct stability analyses on these experiments, by showing the statistical significance of the improvements achieved by
+
+Table 8: Ablation results on DuBIN. RA $( \% )$ on CIFAR10/100-C with WRN40-2 backbone are reported. Mean and standard derivation over three random seeds are shown for each experiment.   
+
+<table><tr><td>Method</td><td>CIFAR10-C</td><td>CIFAR100-C</td></tr><tr><td>AugMix-BN</td><td>89.01 (± 0.03)</td><td>64.56 (± 0.04)</td></tr><tr><td>AugMix-IBN</td><td>89.17 (± 0.24)</td><td>63.94 (± 0.28)</td></tr><tr><td>AugMix-DuBN</td><td>89.11 (± 0.21)</td><td>64.07 (± 0.38)</td></tr><tr><td>AugMix-DuBIN</td><td>89.74(± 0.35)</td><td>65.02 (± 0.58)</td></tr><tr><td>AugMax-DuBN</td><td>89.60 (± 0.62)</td><td>65.06 (± 0.28)</td></tr><tr><td>AugMax-DuBIN</td><td>90.67 (± 0.16)</td><td>66.35 (± 0.21)</td></tr></table>
+
+AugMax-DuBIN over the algorithm randomness. Specifically, we run all experiments in Table 8 using three different random seeds, and report the mean (denoted as $\mu _ { . }$ ) and standard derivation (denoted as $\sigma$ ) of accuracy in the form of $\mu$ $( \pm \sigma )$ in Table 8. As we can observe, the improvements achieved by AugMax-DuBIN are statistically significant and consistent.
+
+AugMax Hyperparameters In this paragraph, we check the sensitivity of AugMax with respect to its hyperparameters: maximization steps $n$ , step size $\alpha$ and consistency loss tradeoff parameter $\lambda$ . The results are shown in Table 9. We first fix $\lambda = 1 0$ and try different values for $n$ and $k$ . We fix $k = 1$ when adjust
+
+Table 9: Ablation results on AugMax-DuBIN update step number $n$ and, early-stopping step $k$ and consistency loss tradeoff parameter $\lambda$ . Results are reported on CIFAR10/CIFAR10-C with WRN40-2.   
+
+<table><tr><td></td><td colspan="2">n</td><td colspan="3"></td><td colspan="3"></td></tr><tr><td></td><td>5</td><td>10</td><td>1</td><td>k2</td><td>3</td><td>1</td><td>入 10</td><td>15</td></tr><tr><td>SA(%)</td><td>95.78</td><td>95.68</td><td>95.68</td><td>95.60</td><td>95.51</td><td>95.90</td><td>95.68</td><td>95.88</td></tr><tr><td>RA (%)</td><td>90.49</td><td>90.67</td><td>90.67</td><td>90.38</td><td>90.21</td><td>88.89</td><td>90.67</td><td>90.30</td></tr></table>
+
+ing $n$ and fix $n = 1 0$ when adjusting $k$ . As we can see, AugMax is stable with respect to different values of $n$ and $k$ . We use $n = 1 0$ and $k = 1$ as the default values since they empirically yield good results at low training overhead. We then fix $n = 1 0 , k = 1$ and adjust $\lambda$ . We find robustness peaks at around $\lambda = 1 0$ , which we set as the default value.
+
+Comparison with Different Diversity and Hardness Strategies In this paragraph, we study different strategies to combine diversity and hardness. The results confirm that diversity and hardness are two complementary dimensions and their proper unification boosts model robustness.
+
+Specifically, we compare AugMax with baseline methods from each strategy group (i.e., diversity or hardness), and other possible strategies to combine both. Besides AugMix and Adversarial Training (e.g., PDGAT [6], FAT [28]), which are the two representative examples for diversity and hardness as shown in Figure 1, we also design the following baseline methods. (1) AugMix+PGDAT: using both AugMix and adversarial images for augmentation. This is a naive baseline to combine diversity with hardness. (2)
+
+Table 10: Results of different augmentation strategies on CIFAR100(-C) with ResNet18.   
+
+<table><tr><td>Method</td><td>Strategy</td><td>SA(%)</td><td>RA(%)</td></tr><tr><td>Normal</td><td>-</td><td>77.99</td><td>48.46</td></tr><tr><td>AugMix [5]</td><td>(diversity)</td><td>78.23</td><td>62.67</td></tr><tr><td>PGDAT [6]</td><td rowspan="4">(hardness)</td><td>60.94</td><td>47.71</td></tr><tr><td>FAT[28]</td><td>61.51</td><td>48.70</td></tr><tr><td>AdvMax</td><td>56.61</td><td>38.65</td></tr><tr><td>AugMix+PGDAT</td><td>61.68</td><td>51.39</td></tr><tr><td>AdvMix</td><td rowspan="2">(diversity &amp;adversity)</td><td>72.36</td><td>56.89</td></tr><tr><td>AugMax-DuBIN</td><td>78.52</td><td>64.02</td></tr></table>
+
+AdvMix: applying adversarial attacks on the augmentation hyperparameters such as the rotation angles and translation pixel numbers [71, 72], while randomly selecting the mixing parameters; (3) AdvMax: applying adversarial attacks on both the augmentation hyperparameters and the mixing weights.5 Please see Appendix B for their implementation details.
+
+The results are shown in Table 10. AugMax-DuBIN achieves the best performance, outperforming all methods from either diversity or hardness group. This shows diversity and hardness to be indeed complementary. On the other hand, the other two naive baselines jointly considering diversity and hardness achieve poorer performance, showing it nontrivial to design a method achieving good balance between diversity and hardness.
+
+# 4.4 Robustness against Other Distribution Shifts
+
+Although AugMax mainly aims at improving model robustness against common corruptions, we find that it can also gain robustness against other types of distribution shifts. Specifically, we evaluate on the distribution shifts caused by the differences in data collection process using CIFAR10.1 [30], and against Spatial Transform adversarial Attacks (STA) [71] on CIFAR10-STA. We generate CIFAR10-STA by applying the worst-of- $k$ attack with $k = 1 0$ on CIFAR10 test set following [71].
+
+Table 11: Robustness against other distribution shifts. Accuracy $( \% )$ on CIFAR10.1 and CIFAR10-STA are evaluated on ResNeXt29 trained on CIFAR10.   
+
+<table><tr><td>Method</td><td>CIFAR10.1</td><td>CIFAR10-STA</td></tr><tr><td>Normal</td><td>88.90</td><td>30.30</td></tr><tr><td>AugMix</td><td>88.90</td><td>54.30</td></tr><tr><td>AugMax-DuBIN</td><td>90.64</td><td>63.20</td></tr></table>
+
+Results are reported in Table 11 (using the same ResNeXt29 models trained on CIFAR10 as in Section 4.2). AugMax-DuBIN steadily outperforms AugMix against both distributional shifts.
+
+# 5 Conclusion
+
+In this paper, we propose AugMax, a novel data augmentation strategy, that significantly improves robustness by strategically unifying diversity and hardness. To enable efficient training while facing the resulting heterogeneous features, we design a novel normalization scheme termed DuBIN. Using the combination of AugMax and DuBIN, we consistently demonstrate state-of-the-art robustness on several natural corruption benchmarks.
+
+AugMax unifies diversity and hardness in a heuristic manner. While we showed in ablation studies that AugMax outperforms other heuristic combinations, there may be better trade-offs available. We leave such study for future work and focus in this paper on showing the benefit of unifying diversity and hardness, which were separately considered in previous research.
+
+# Acknowledgement
+
+Z.W. is supported by the U.S. Army Research Laboratory Cooperative Research Agreement W911NF17-2-0196 (IOBT REIGN), and an NVIDIA Applied Research Accelerator Program.
+
+# References
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+    {
+        "type": "text",
+        "text": "AugMax: Adversarial Composition of Random Augmentations for Robust Training ",
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+        "text": "Haotao Wang1∗, Chaowei $\\mathbf { X i a o ^ { 2 , 3 } }$ , Jean Kossaifi2, Zhiding $\\mathbf { Y } \\mathbf { u } ^ { 2 }$ , Anima Anandkumar2,4, and Zhangyang Wang1 ",
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+        "text": "1Department of Electrical and Computer Engineering, University of Texas at Austin 2NVIDIA 3Arizona State University 4California Institute of Technology 1{htwang, atlaswang}@utexas.edu 2{chaoweix, jkossaifi, zhidingy, aanandkumar}@nvidia.com ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Data augmentation is a simple yet effective way to improve the robustness of deep neural networks (DNNs). Diversity and hardness are two complementary dimensions of data augmentation to achieve robustness. For example, AugMix explores random compositions of a diverse set of augmentations to enhance broader coverage, while adversarial training generates adversarially hard samples to spot the weakness. Motivated by this, we propose a data augmentation framework, termed AugMax, to unify the two aspects of diversity and hardness. AugMax first randomly samples multiple augmentation operators and then learns an adversarial mixture of the selected operators. Being a stronger form of data augmentation, AugMax leads to a significantly augmented input distribution which makes model training more challenging. To solve this problem, we further design a disentangled normalization module, termed DuBIN (Dual-Batch-and-Instance Normalization), that disentangles the instance-wise feature heterogeneity arising from AugMax. Experiments show that AugMax-DuBIN leads to significantly improved out-of-distribution robustness, outperforming prior arts by $3 . 0 3 \\%$ , $3 . 4 9 \\%$ , $1 . 8 2 \\%$ and $0 . 7 1 \\%$ on CIFAR10-C, CIFAR100-C, Tiny ImageNet-C and ImageNet-C. Codes and pretrained models are available: https://github.com/VITA-Group/AugMax. ",
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+        "type": "text",
+        "text": "1 Introduction ",
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+        "type": "text",
+        "text": "Out-of-distribution (OOD) samples present a challenge when deploying AI models in the real world. Examples include natural corruptions (e.g., due to camera blurs or noise, snow, rain, or fog in image data), sensory perturbations (e.g., sensor transient error, electromagnetic interference) and domain shifts (e.g., summer winter). However, deep networks are often trained on limited amounts of data which may not cover sufficient scenarios. As a result, they are vulnerable to unforeseen distributional changes despite achieving high performance on standard benchmarks [1–3]. This jeopardizes their trustworthiness as well as safe deployment in real-world environments. Thus it is critical to develop techniques that improve robustness even when training with relatively clean datasets. ",
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+        "text": "Several techniques have been proposed to consolidate the robustness against unforeseen corruptions, including robust data augmentation [5, 3, 8, 6], Lipschitz continuity [9–11], stability training [12], pre-training [13–16], and robust network structures [17–19], to name a few. Among these techniques, data augmentation is of particular interest due to its empirical effectiveness, ease of implementation, low computational overhead, and plug-and-play nature. ",
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+        "img_path": "images/96da970a0935f651f016706faf1673e344b89a5c13bdb2ecde02923b0a6543ea.jpg",
+        "image_caption": [
+            "Figure 1: The effects of diversity and hardness in data augmentations. We visualize features of augmented images fed to the network during training. The features are from the penultimate layer of a ResNeXt29 trained on CIFAR10 training data using only standard data augmentation (random flipping and translation) following [4]. For visualization, we randomly selected 300 fixed images from 3 fixed classes (denoted by different colors) to which we apply different augmentation methods: (a) standard augmentation (random flipping and translation); (b) AugMix [5]; (c) PGD Attack [6, 7]; (d) AugMax (ours). In order to achieve good model robustness, the augmented training data should both be diverse and also contain enough hard cases. As can be seen, PGD Attack generates hard cases (which the network cannot separate) but are not diverse enough (they are all clustered together) while AugMix creates diverse but not hard samples (they are well separated). By contrast, our approach (AugMax) achieves a unification between hard and diverse samples. "
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+        "text": "There are mainly two categories of data augmentation approaches: ",
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+    {
+        "type": "text",
+        "text": "The first category aims to increase diversity of the training data by composing multiple random transformations [20–22, 5, 3]. While standard data augmentation methods (e.g., random flipping and translation) lead to poor robustness [5], more aggressive combinations of multiple augmentations have shown promise. One such successful example is AugMix [5], which stochastically samples from diverse augmentation operations and randomly mixing them to produce highly diverse augmented images. AugMix can increase the sample diversity coverage compared to standard augmentations, as shown in Figure 1 (a) and (b). ",
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+    {
+        "type": "text",
+        "text": "The second category aims to boost the hardness of the training data by sampling from the worst-case augmentations that tries to fool the model into misclassifying the samples. A common technique to achieve this goal is adversarial perturbation [6, 23, 8]. Training over such worst-case samples allows a model to actively fix its generalization weaknesses [24] and empirically improves its robustness against corruptions [25, 26, 7]. Due to the extra complexity to generate adversarial perturbations, the improved robustness usually comes at the cost of largely increased training time compared with non-adversarial methods [6, 27]. An example falling into this category is the PGD attack [6], as is shown in Figure 1 (c). ",
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+    },
+    {
+        "type": "text",
+        "text": "Previous work has focused on leveraging one of these category to improve robustness. In this paper, we unify both approaches in a single framework. In particular, we show that diversity and hardness are complementary and that a unification between the two is necessary to achieve robustness. We propose a new strategy to achieve this unification and successfully increase robustness. ",
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+    {
+        "type": "text",
+        "text": "Summary of contributions: ",
+        "text_level": 1,
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+        "text": "• We propose AugMax, a novel augmentation framework which achieves robustness through a unification between diversity and hardness, by searching for the worst-case mixing strategy. • Being a stronger form of data augmentation, AugMax leads to a significantly augmented and more heterogeneous input distribution, which also makes model training more challenging. To solve this problem, we design a new normalization strategy, termed DuBIN, to disentangle the instance-wise feature heterogeneity of AugMax samples. • We show that combination of AugMax and DuBIN (AugMax-DuBIN) achieves state-of-the-art robustness against corruptions and improves robustness against other common distribution shifts. ",
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+    {
+        "type": "text",
+        "text": "AugMax achieves a good unification between sample diversity and hard corner-case generation during data augmentation. AugMax is built on top of the AugMix framework [5] which mixes multiple data augmentation operators in a multi-branch and layered pipeline. However, different from AugMix where augmentation operators and mixing weights are both randomly sampled, operators are first randomly sampled, followed by adversarially trained mixture of the selected operators in Augmax. This simple change from AugMix to AugMax results in considerable difference in their feature distributions. From the visualizations (in Figure 1), AugMax generates more adversarially “hard samples\", while still keeping a good amount of diversity compared to AugMix and PGD Attack. Searching for adversarial mixing strategies in AugMax is slightly more expensive since it involves adversarial mixing. To make this efficient, we adopt an efficient adversarial training strategy [28]. As a result, AugMax adds only a reasonable amount of extra complexity compared to non-adversarial training methods, while improving robustness significantly. For example, AugMax training time is only $\\sim 1 . 5$ times that of AugMix on ImageNet (see Table 7). Moreover, Augmax is significantly more efficient than traditional adversarial training which has $\\sim 1 0$ times training time overhead compared with AugMix. ",
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+    {
+        "type": "image",
+        "img_path": "images/5c2b2929baf2ee2ecdb10d564875ad242f4ebe5bd3f8d0fcce5c19740424a479.jpg",
+        "image_caption": [
+            "Figure 2: AugMax overview. Black and red arrows represent forward paths and back-propagation paths to generate AugMax images, respectively. In contrast to AugMax, where the mixing parameters $m$ and $\\pmb { w }$ are adversarially learned, AugMix randomly samples $m$ and $\\pmb { w }$ from predefined distributions (and thus no backpropergation on $m$ and $\\textbf { \\em w }$ ). "
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+    {
+        "type": "text",
+        "text": "Being a stronger form of data augmentation with adversarial sample generation, AugMax leads to a significantly augmented input distribution which makes model training more challenging. This naturally motivates us to propose a novel and finer-grained normalization scheme termed Dual-Batchand-Instance Normalization (DuBIN). As illustrated in Figure 3, DuBIN adds an additional instance normalization (IN) in parallel to the traditional Dual Batch Normalization (DuBN) [7, 29] used in adversarial training, in order to better model and disentangle the instance-wise feature heterogeneity arising from AugMax. We show that adding the instance normalization is an important knob to promote the capability of AugMax in boosting model robustness. ",
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+    {
+        "type": "text",
+        "text": "Our framework, AugMax-DuBIN, is illustrated in Figure 2. AugMax-DuBIN trains on clean images and achieves state-of-the-art robustness on natural corruption benchmarks [1], and also improves model robustness against other common distribution shifts [30, 2]. In particular, our method surpasses state-of-the-art method on CIFAR10-C, CIFAR100-C, Tiny ImageNet-C and ImageNet-C by $3 . 0 3 \\%$ , $3 . 4 9 \\%$ , $1 . 8 2 \\%$ and $0 . 7 1 \\%$ respectively. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.1 Robustness to Distributional Shifts ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hendrycks et al. [1] pioneer the study of prediction errors exhibited by machine learning models on unseen natural corruptions and perturbations. They introduce two variants of the original ImageNet validation set to evaluate the model’s robustness: the ImageNet-C for input corruption robustness, and the ImageNet-P for input perturbation robustness. The former applies 15 diverse corruptions drawn from four main categories (noise, blur, weather, and digital) on ImageNet validation images. The latter contains perturbation sequences generated from each image. Recht et al. [30, 2] collect CIFAR10.1 and ImageNet-V2 as a reproduction of the original CIFAR10 [31] and ImageNet [32] test sets, and find that a minute natural distribution shift caused by the minutiae in dataset collection process leads to a large drop in accuracy for a broad range of image classifiers. Geirhos et al. [33] observe that deep models trained on ImageNet are biased towards textures, and the model robustness can be improved by emphasizing more on global shape features [33–36]. Wang et al. [37] find that the benchmark test performance does not always translate to the real-world generalizability, and propose an efficient method to troubleshoot trained models using real-world unlabeled images. Hendrycks et al. [38] collect ImageNet-A with the new concept of natural adversarial examples: unmodified real-world images that falsify common machine learning models. Other representative benchmark efforts include [39–42, 3], all demonstrating the brittleness of deep models under various distribution shifts. Readers of interest are referred to a recent survey [43]. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "2.2 Data Augmentation: Random and Adversarial ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Data augmentations have been widely used to increase training set diversity and improve model generalization ability [44, 21, 33, 22]. Among those methods, AugMix [5] achieves the outstanding robustness against natural corruptions by randomly mixing multiple diverse augmentations. It establishes high performance bars on many natural corruption benchmarks including CIFAR10-C, CIFAR100-C and ImageNet-C [1]. DeepAugment [3] feeds clean images to image-to-image models with randomly perturbed model weights to generate visually diverse augmented images. Combined with AugMix, it achieves state-of-the-art robustness on ImageNet-C [1]. Lately, a concurrent work MaxUp [45] generates hard training samples by selecting the worst-case weights in the MixUp [20] pipeline, to improve both model generalization ability and adversarial robustness. Our work shared a similar mindset to MaxUp, but we build on the more sophisticated AugMix pipeline and handle the resulting higher feature heterogeneity using a new normalization module. ",
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+    {
+        "type": "text",
+        "text": "Besides, Adversarial training (AT) [6, 46–52] utilizes adversarial samples as a special data augmentation method, and has also shown promise in improving model robustness against natural corruptions [23] and domain gaps [25], usually at the cost of increasing training time [27] and degrading the standard accuracy on clean images [53]. Volpi et al. [25] and Zhao et al. [26] augment the training set with samples from a fictitious adversarial domain to improve domain adaptation performance. Xie et al. [7] show that adversarial training can improve both standard accuracy and robustness against distribution shifts, with the help of auxiliary batch normalization. Adversarial Noise Training (ANT) [8] uses random Gaussian noise with learned adversarial hyperparameters (mean and variance) as the augmentation. Gowal et al. [54] propose to generate new augmented images by adversarially composing the representations of different original images. Wang et al. [55] improve the robustness of pose estimation models by adversarially mixing different corrupted images as data augmentation. Based on DeepAugment [3], Calian et al. [56] propose AdversarialAugment, which optimizes the parameters of image-to-image models to generate adversarially augmented images, achieving promising results against natural corruptions. Robey et al. [57] proposed model-based robust learning which generates augmented images with adversarial natural variations to improve model robustness against naturally-occurring conditions (e.g., snow, decolorization, shadow, and many others). The authors further generalized such idea to the domain generalization problem and achieved impressive results [58]. ",
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+    {
+        "type": "text",
+        "text": "2.3 Normalization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Batch Normalization (BN) [59] has been a cornerstone for training deep networks. Inspired by BN, more task-specific modifications are proposed by exploiting different normalization axes, such as Instance Normalization (IN) [60], Layer Normalization (LN) [61] and Group Normalization (GN) [62]. A few latest works investigate to use multiple normalization layers instead of one BN. Zajkac et al. [63] use two separate BNs to handle the domain shift between labeled and unlabeled data in semi-supervised learning. Xie et al. [64, 7] observe the difference between standard and adversarial feature statistics during AT, and craft dual batch normalization (DuBN) to disentangle the standard and adversarial feature statistic to improve both the standard accuracy and robustness. While most deep models just use one normalization type throughout the network, IBN-Net [29] unifies IN and BN in one model. It finds that when applied together, IN tends to learn features invariant to appearance (colors, styles, etc.) changes, while BN is essential for preserving high-level semantic contents. IBN-Net shows that combining IN and BN in an appropriate manner can improve model generalization. A similar combination was also explored in the style transfer field [65], learning to selectively normalize only disturbing styles while preserving useful styles. ",
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+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Preliminary: The AugMix Framework ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "At a high level, AugMix creates a new composite image from a clean sample by first applying several simple augmentation operations to it before combining the results through random linear combinations. This augmentation scheme is coupled with a Jensen-Shannon consistency loss that enforces feature similarity among clean and augmented images. Specifically, AugMix consists of multiple augmentation chains (3 by default), each applied in parallel to the same input image. Each chain composes one to three randomly selected augmentation operations. These augmentations are the same as AutoAugment [21], but excluding those that overlap with ImageNet-C corruptions. The augmented images from each augmentation chains are then combined together with the original, clean sample using a linear combinations with randomly samples coefficients. The final image thus incorporates several sources of randomness, coming respectively from the choice of operations, the severity level of these operations, the lengths of the augmentation chains and the mixing weights. Despite its simplicity, AugMix achieves state-of-the-art corruption robustness. In the remaining of this section, we build on the previous AugMix work and propose AugMax, an augmentation strategy that further improves robustness. ",
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+        "type": "text",
+        "text": "3.2 AugMax: Augmented Training with Unified Diversity and Hardness ",
+        "text_level": 1,
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+        "text": "In this section, we rigorously introduce our new data augmentation method, AugMax, illustrated in Figure 2. At a high level, the main difference between AugMax and AugMix is a new optimization procedure used to select the mixing weights $m$ and $\\pmb { w }$ . ",
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+        "type": "text",
+        "text": "Given a data distribution $\\mathbb { D }$ over images $\\pmb { x } \\in \\mathbb { R } ^ { d }$ and labels $\\pmb { y } \\in \\mathbb { R } ^ { c }$ , our goal is to train a mapping (classifier) $f : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { c }$ from images to output softmax probabilities, parameterized by $\\pmb \\theta$ , that is robust to (unknown) distribution shifts. We minimize the empirical risk ",
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+    {
+        "type": "equation",
+        "img_path": "images/a631f8158e3c3510a2b1778b286610a810777e1ae37735657adb9f940a9de580.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pmb { \\theta } } \\mathbb { E } _ { ( \\pmb { x } , \\pmb { y } ) \\sim \\mathcal { D } } \\ \\mathcal { L } ( f ( \\pmb { x } ; \\pmb { \\theta } ) , \\pmb { y } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal L ( \\cdot , \\cdot )$ is the loss function (e.g., cross-entropy). As illustrated in Figure 2, an AugMax image $\\pmb { x } ^ { * }$ is generated by learning a set of adversarial mixing parameters $m ^ { * } , w ^ { * }$ which maximizes the loss: ",
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+    {
+        "type": "equation",
+        "img_path": "images/14101d09aba4d80f0781989e97d50b52c7a08fe238361310614c6cf7955355bc.jpg",
+        "text": "$$\n{ \\pmb x } ^ { * } = g ( { \\pmb x } _ { o r i g } ; m ^ { * } , { \\pmb w } ^ { * } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $g ( \\cdot )$ denotes the AugMax augmentation function, $\\pmb { x } _ { o r i g }$ is the original image, and ",
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+    {
+        "type": "equation",
+        "img_path": "images/c23a97a37d79c305bc7020c80779d9ac167706189ff9b52cb1bf591e07cbdc5f.jpg",
+        "text": "$$\nm ^ { * } , \\boldsymbol { w } ^ { * } = \\underset { m , w } { \\arg \\operatorname* { m a x } } \\mathcal { L } \\big ( f ( g ( x _ { o r i g } ; m , w ) ; \\theta ) , y \\big ) , \\quad \\mathrm { s . t . } m \\in [ 0 , 1 ] , w \\in [ 0 , 1 ] ^ { b } , w ^ { T } \\mathbf { 1 } = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To simplify the optimization problem in Eq. (3), we use a re-parameterization trick by setting ${ \\pmb w } = \\sigma ( { \\pmb p } )$ , where $\\sigma ( \\cdot )$ is the softmax function, and convert the optimization into a differentiable one: ",
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+    {
+        "type": "equation",
+        "img_path": "images/9de6ffc90bbf1f5c1267b912d938465cd47e60f4aa4d7946de41bf4a48ca042c.jpg",
+        "text": "$$\nm ^ { * } , \\pmb { p } ^ { * } = \\operatorname * { a r g m a x } _ { m \\in [ 0 , 1 ] , \\pmb { p } \\in \\mathbb { R } ^ { b } } \\mathcal { L } ( f ( g ( \\pmb { x } _ { o r i g } ; m , \\sigma ( \\pmb { p } ) ) ; \\theta ) , \\pmb { y } ) ; \\quad \\pmb { w } ^ { * } = \\sigma ( \\pmb { p } ^ { * } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training with AugMax augmentation can then be written altogether as minimax optimization: ",
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+    {
+        "type": "equation",
+        "img_path": "images/3ef15a396215c1497c3bd04ec00fdd967dcb0a10a41659854538f07087ba2cc7.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { \\theta } { \\operatorname* { m i n } } \\mathbb { E } _ { ( \\boldsymbol { x } , \\boldsymbol { y } ) \\sim \\mathcal { D } } \\frac { 1 } { 2 } [ \\mathcal { L } ( f ( \\boldsymbol { x } ^ { * } ) ; \\boldsymbol { \\theta } ) , \\boldsymbol { y } ) + \\mathcal { L } ( f ( \\boldsymbol { x } ) ; \\boldsymbol { \\theta } ) , \\boldsymbol { y } ) ] + \\lambda \\mathcal { L } _ { c } ( \\boldsymbol { x } , \\boldsymbol { x } ^ { * } ) } \\\\ & { \\mathrm { s . t . } \\boldsymbol { x } ^ { * } = g ( \\boldsymbol { x } ; m ^ { * } , \\boldsymbol { w } ^ { * } ) ; \\boldsymbol { w } ^ { * } = \\sigma ( p ^ { * } ) ; \\boldsymbol { m } ^ { * } , p ^ { * } = \\underset { m \\in [ 0 , 1 ] , p \\in \\mathbb { R } ^ { b } } { \\arg \\operatorname* { m a x } } \\mathcal { L } ( f ( g ( \\boldsymbol { x } ; m , \\sigma ( p ) ) ; \\boldsymbol { \\theta } ) , \\boldsymbol { y } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { c }$ is a consistency loss regularizing augmented images to have similar model outputs with the original images and $\\lambda$ is the trade-off parameter. Our implementation of $\\mathcal { L } _ { c }$ is adapted from [5]: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5c2db52eeee70ffef2952f6a0943d11f39eae03ed2a7c376b278502b91f23a3.jpg",
+        "text": "$$\n\\mathcal { L } _ { c } ( { \\pmb x } , { \\pmb x } ^ { * } ) = \\mathrm { J S } ( f ( { \\pmb x } ; \\theta ) , f ( \\tilde { { \\pmb x } } ; \\theta ) , f ( { \\pmb x } ^ { * } ; \\theta ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $\\mathrm { J } \\mathrm { S } ( \\cdot )$ being the Jensen-Shannon divergence and $\\tilde { \\pmb x }$ being the augmented image generated by AugMix from $_ { \\textbf { \\em x } }$ . In order to reduce the training complexity overhead caused by adversarial training, we employ an accelerated adversarial attack method [28] to solve Eq. (4), which adds only a reasonable amount of extra complexity compared to AugMix baseline. The algorithm to solve Eq. (5) is summarized in Appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Visualization of the effects of diversity and hardness To illustrate the motivation behind AugMax, we visualize the feature representations induced by different augmentation methods to showcase their diversity and hardness in Figure 1. Specifically, we visualize the impact of various data augmentation methods on the feature representations obtained with a ResNeXt29 that is normally trained on CIFAR-10. Note that the model is trained using only default standard augmentations (i.e., random flipping and translation), and therefore has never seen augmentation from AugMix, AugMax, or PGD attack. Following [4], we visualize the penultimate layer feature distributions of training samples from three different classes. Three hundred training images are selected from each class for visualization. For AugMix and $\\ell _ { \\infty }$ PGD attack, we use the same hyperparameters in the original papers [5, 6]. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We observe in Figure 1 that the clean images form well-separated clusters in the feature space, leaving large regions between those clusters empty. This potentially causes uncertain and unreliable generalization if new test samples (from a shifted distribution) fall into those regions. AugMix enlarges each cluster to cover a broader region, effectively increasing sample diversity. However, after augmentation, very few samples are close to the decision boundaries, revealing an inability to generate hard samples. Samples from the PGD attack, on the other hand, collapse into a small region around the classification borders, losing feature diversity. We further investigate the method through ablation studies in Section 4.3. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 DuBIN: Disentangled Normalization for Heterogeneous Features ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "AugMax leads to better coverage of the input space by unifying sample diversity and hardness, as demonstrated in Figure 1. While such more versatile distribution in data augmentation has the potential to increase robustness, we do observe that naive incorporation of AugMax leads to relatively marginal improvement over AugMix (see Section 4.3). This is due to the comprehensiveness of AugMax can also lead to a higher feature heterogeneity, which may require larger model capacity to encode. To address this problem, we design a novel normalization layer, termed Dual Batch-and-Instance Normalization (DuBIN), to disentangle the instance-level heterogeneity. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Figure 3, DuBIN consists of two parallel parts: (1) a dual batch normalization (DuBN) [7], which is the by-default normalization layer to use for adversarial training [7, 64], to disentangle the group-level statistics of the clean and corner-case augmented samples2; and (2) an additional instance normalization, to account for instancelevel feature statistics due to augmentation diversity. At each layer, the input feature is split into halves along the channel dimension: one half is then fed into IN, and the other fed into DuBN. Finally, the outputs of IN and DuBN are concatenated back along the channel dimension for the next layer. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/37442dadfb77175a367b93c847e84dc9a4c01bb733ad3830171f67604343a360.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of DuBIN. “Split” and “cat” represent splitting and concatenating on features along the channel dimension. The input feature $x$ with $C$ channels is split into halves (i.e., $x _ { A }$ and $x _ { B }$ , each with $C / 2$ channels). Each half goes into different normalization layers (either IN or DuBN) and the outputs are concatenated back along the channel dimension. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To provide additional insights of DuBIN, we compare the BN statistics of AugMax-DuBN and AugMax-DuBIN in Table 1. $\\overline { { \\sigma } } _ { c } ^ { 2 }$ and $\\overline { { \\sigma } } _ { a } ^ { 2 }$ represent the average batch normalization variance over all channels of $B N _ { c }$ and $B N _ { a }$ , respectively. We observe that both $\\overline { { \\sigma } } _ { c } ^ { 2 }$ and $\\overline { { \\sigma } } _ { a } ^ { 2 }$ are smaller in the DuBIN network than those in the DuBN counterpart. This shows that the IN branch in DuBIN can reduce the feature diversity that BN otherwise needs to model, by ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/6521c0c4244f61bde72b4214dc84b4deb685039d7276a902a8bc5bb391b8d53f.jpg",
+        "table_caption": [
+            "Table 1: BN statistics of different layers in WRN40-2 with DuBN or DuBIN trained on CIFAR100 using AugMax training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Layer</td><td>Block 1 layer 2</td><td>Block1 layer 3</td><td>Block 2 layer 2</td></tr><tr><td rowspan=\"3\">AugMax-DuBN</td><td></td><td>0.0369</td><td>0.0450</td><td>0.0301</td></tr><tr><td>品</td><td>0.0469</td><td>0.0585</td><td>0.0382</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">AugMax-DuBIN</td><td></td><td>0.0306</td><td>0.0403</td><td>0.0264</td></tr><tr><td></td><td>0.0348</td><td>0.0466</td><td>0.0292</td></tr><tr><td>R</td><td></td><td></td><td></td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "encoding instance-level diversities. With lower feature variation, BNs can converge better with improved performance [66]. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here, we introduce in detail the experimental setting, and report quantitative results for robustness against natural corruptions and distribution shifts. We show that our method induces better robustness than existing ones. We study the properties of our approach in thorough ablation studies. ",
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+    {
+        "type": "text",
+        "text": "4.1 Experimental Setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Datasets and Models We evaluate our proposed method on CIFAR10, CIFAR100 [31], ImageNet [32] and Tiny ImageNet $\\left( \\mathrm { T I N } \\right)$ . For neural architectures, we use a ResNet18 [67], WRN40-2 [68] and ResNeXt29 [69] on the CIFAR datasets, and ResNet18 on ImageNet and Tiny ImageNet4. To evaluate the model’s robustness against common natural corruptions, we use CIFAR10-C, CIFAR100- C, ImageNet-C and Tiny ImageNet-C (TIN-C) [1], which are generated by adding natural corruptions to the original test set images. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics Following established conventions [1, 5], we define robustness accuracy (RA) as the average classification accuracy over all 15 corruptions, to measure the models’ robustness on CIFAR10-C and CIFAR100-C. On ImageNet-C and Tiny ImageNet-C, we use both RA and mean corruption error (mCE) to evaluate robustness. As defined by [1], mCE is the weighted average of target model corruption ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/65bb5956d15efb8978beddf15b068abe3652c3115f51fdc8b7dc6d813d37da55.jpg",
+        "table_caption": [
+            "Table 2: Evaluation results on CIFAR10 and CIFAR10-C. The best metric is shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Metric</td><td>Normal</td><td>AugMix</td><td>AugMax-DuBIN</td></tr><tr><td rowspan=\"2\">ResNet18</td><td>SA(%)</td><td>95.56</td><td>95.79</td><td>95.76 (-0.03)</td></tr><tr><td>RA(%)</td><td>74.75</td><td>89.49</td><td>90.36 (+0.87)</td></tr><tr><td rowspan=\"2\">WRN40-2</td><td>SA(%)</td><td>94.78</td><td>95.67</td><td>95.68 (+0.01)</td></tr><tr><td>RA(%)</td><td>73.71</td><td>89.01</td><td>90.67 (+1.66)</td></tr><tr><td rowspan=\"2\">ResNeXt29</td><td>SA(%)</td><td>95.60</td><td>96.25</td><td>96.39 (+0.14)</td></tr><tr><td>RA(%)</td><td>71.70</td><td>89.08</td><td>92.11 (+3.03)</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "errors normalized by the corruption errors of a baseline model across different types of corruptions. We use the AlexNet provide in [1] as the baseline model for experiments on ImageNet. Since [1] does not provide an off-the-shelf baseline model for Tiny ImageNet, we use the conventionally trained ResNet18 on Tiny ImageNet as the baseline model for experiments on Tiny ImageNet. We use standard accuracy (SA) to denote the classification accuracy on the original clean testing images. ",
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+    {
+        "type": "text",
+        "text": "Baseline Methods On CIFAR10-C and CIFAR100-C, we compare our method with the state-of-the-art method, AugMix [5]. In addition, AugMix has previously been combined with other orthogonal methods, such as DeepAugment [3], to further boost performance on ImageNet-C. Specifically, [3] combine DeepAugment as an orthogonal component to AugMix, and achieve state-of-the-art ",
+        "bbox": [
+            174,
+            517,
+            485,
+            640
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b95f8172541a3329b5d891c37cf4a09e10607f97bb7aa767e9154c73614f9fc4.jpg",
+        "table_caption": [
+            "Table 3: Evaluation results on CIFAR100 and CIFAR100-C. The best metric is shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Metric</td><td>Normal</td><td>AugMix</td><td>AugMax-DuBIN</td></tr><tr><td rowspan=\"2\">ResNet18</td><td>SA(%)</td><td>77.99</td><td>78.23</td><td>78.69 (+0.46)</td></tr><tr><td>RA(%)</td><td>48.46</td><td>62.67</td><td>65.75 (+3.08)</td></tr><tr><td rowspan=\"2\">WRN40-2</td><td>SA(%)</td><td>76.19</td><td>77.03</td><td>76.80 (-0.23)</td></tr><tr><td>RA(%)</td><td>46.80</td><td>64.56</td><td>66.35 (+1.79)</td></tr><tr><td rowspan=\"2\">ResNeXt29</td><td>SA(%)</td><td>79.95</td><td>78.58</td><td>80.70 (+2.12)</td></tr><tr><td>RA(%)</td><td>47.76</td><td>65.37</td><td>68.86 (+3.49)</td></tr></table>",
+        "bbox": [
+            498,
+            546,
+            821,
+            637
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "performance on ImageNet-C with DeepAugment $^ +$ AugMix [3]. To allow comparison with this scenario, we also run the following comparison experiment on ImageNet-C and Tiny ImageNet-C: (1) AugMix v.s. AugMax and (2) DeepAugment $^ +$ AugMix v.s. DeepAugment $^ +$ AugMax. We show that the advantages of AugMax still exist when combined with other orthogonal methods. ",
+        "bbox": [
+            173,
+            640,
+            826,
+            695
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we also compare AugMax with ANT [8], which utilizes Gaussian noises with learned distribution parameters for data augmentation. Since models tend to generalize better on test images similar to those seen during training [70], ANT mainly improves robustness against high frequency corruptions (e.g., additive noises), while achieving comparable robustness on other unseen corruptions (e.g., bluring) with AugMix. Following [8], we perform the comparison on the subset of Tiny ImageNet-C with 12 remaining corruptions after removing all additive noises. ",
+        "bbox": [
+            174,
+            703,
+            485,
+            854
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/aabc68e64ce968bfe73b0d7c75b02f10b41f86800ca9aed521d7ba7a60e61bd4.jpg",
+        "table_caption": [
+            "Table 4: Evaluation results on ImageNet and ImageNet-C. The best metric is shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SA (%,↑)</td><td>RA (%,↑)</td><td>mCE(%,↓)</td></tr><tr><td>Normal</td><td>69.83</td><td>30.91</td><td>87.47</td></tr><tr><td>AugMix</td><td>68.06 67.62</td><td>34.58 35.01</td><td>83.08 82.56</td></tr><tr><td>AugMax-DuBIN</td><td>(-0.44)</td><td>(+0.43)</td><td>(-0.52)</td></tr><tr><td>DeepAugment + AugMix</td><td>65.32</td><td>45.84</td><td>69.29</td></tr><tr><td>DeepAugment +</td><td>64.43</td><td>46.55</td><td>68.47</td></tr><tr><td>AugMax-DuBIN</td><td>(-0.89)</td><td>(+0.71)</td><td>(-0.82)</td></tr></table>",
+        "bbox": [
+            500,
+            734,
+            820,
+            854
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            854,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details We follow [5] for hyperparameter settings. Specifically, for all experiments on CIAFR10 and CIFAR100, we use SGD optimizer with initial learning rate 0.1 and cosine annealing learning rate scheduler, and train all models for 200 epochs. For all experiments on ImageNet, we use SGD optimizer with initial learning rate 0.1 and batch size 256 to train the model for 90 epochs. We reduce the learning rate by $1 / 1 0$ at the 30-th and 60-th epoch. For all experiments on Tiny ImageNet, we train for 200 epochs and decay the learning rate at the 100-th and 150-th epoch. Other hyperparameter settings are identical as those in ImageNet experiments. ",
+        "bbox": [
+            173,
+            92,
+            826,
+            189
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We set batch size to 256 for all experiments. In Eq. (4), $m$ and $\\pmb { p }$ are uniformly randomly initialized between 0 and 1. We use PGD to update $m$ : first do gradient ascend on $m$ and then project it back into the $[ 0 , 1 ]$ interval. $\\pmb { p }$ is updated using gradient ascend. AugMax follows the same rules as AugMix [5] to randomly select augmentation operation type, severity level and chain length. Following [6], we iteratively solve the inner maximization and outer minimization problems in Eq. (5): the inner maximization is updated for $n$ steps, for every 1 step update on ",
+        "bbox": [
+            174,
+            189,
+            483,
+            353
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/501ae04013f4a1903784d4a1135fff609e7484b2c4279b01c47d8716885a1d0e.jpg",
+        "table_caption": [
+            "Table 5: Evaluation results on TIN and TIN-C. The best metric is shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>SA (%,↑)</td><td>RA (%,↑)</td><td>mCE(%,↓)</td></tr><tr><td>Normal</td><td>61.64</td><td>23.91</td><td>100.00</td></tr><tr><td>AugMix</td><td>61.79 62.21</td><td>36.85 38.67</td><td>83.04 80.72</td></tr><tr><td>AugMax-DuBIN</td><td>(+0.42)</td><td>(+1.82)</td><td>(-2.32)</td></tr><tr><td>DeepAugment + AugMix</td><td>59.59</td><td>40.67</td><td>78.28</td></tr><tr><td>DeepAugment +</td><td>59.72</td><td>40.99</td><td>77.83</td></tr><tr><td>AugMax-DuBIN</td><td>(+0.13)</td><td>(+0.32)</td><td>(-0.45)</td></tr></table>",
+        "bbox": [
+            500,
+            224,
+            821,
+            343
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "the outer minimization. Both $m$ and $\\pmb { p }$ are updated with step size $\\alpha = 0 . 1$ . We set $n = 5$ on ImageNet for efficiency and $n = 1 0$ on other datasets. We set $\\lambda$ in Eq. (5) to be 12 on ImageNet and 10 on all other datasets, except for ResNet18 on CIFAR100 where we find $\\lambda = 1$ leads to better performance. All experiments are conducted on a server with four NVIDIA RTX A6000 GPUs. ",
+        "bbox": [
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+            353,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 Robustness against Natural Corruptions ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            428,
+            493,
+            443
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on CIFAR10-C and CIFAR100-C are shown in Table 2 and 3 respectively, where “normal” means training with default standard augmentations (e.g., random flipping and translation). From the results, we see that AugMax-DuBIN achieves new state-of-the-art performance on both datasets with different model structures. For example, compared with AugMix, our method improves accuracy by as high as $3 . 0 3 \\%$ and $3 . 4 9 \\%$ on CIFAR10-C and CIFAR100-C, respectively. Moreover, we observe that our method benefits larger models more. Specifically, in both CIFAR10 and CIFAR100 experiments, the largest robustness gain is achieved on ResNeXt29, which has the largest capacity among the three models. ",
+        "bbox": [
+            173,
+            453,
+            826,
+            564
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on ImageNet are shown in Table 4. AugMax-DuBIN outperforms AugMix by $0 . 5 2 \\%$ in terms of $\\mathrm { m C E }$ on ImageNet-C. Further combining AugMax-DuBIN with DeepAugment outperforms AugMix $^ +$ DeepAugment by $0 . 8 2 \\%$ in terms of mCE, achieving a new stateof-the-art performance on ImageNet-C. This ",
+        "bbox": [
+            174,
+            570,
+            485,
+            667
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4eafb5798d3f48d4f9bff4f996da476789a3f65abf7e46f49e25b2abb52d6303.jpg",
+        "table_caption": [
+            "Table 6: Evaluation results on TIN and TIN-C (w/o noise). The best metric is shown in bold. "
+        ],
+        "table_footnote": [
+            "shows that AugMax can be used as a more advanced basic building block for model robustness and inspires future researches to build other defense methods on top of it. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>SA(%,↑)</td><td>RA(%,↑)</td><td>mCE(%,）</td></tr><tr><td>Normal</td><td>61.64</td><td>24.38</td><td>100.00</td></tr><tr><td>AugMix</td><td>61.79</td><td>37.63</td><td>82.54</td></tr><tr><td>ANT</td><td>61.26</td><td>35.30</td><td>85.70</td></tr><tr><td>AugMax-DuBIN</td><td>62.21</td><td>38.66</td><td>80.29</td></tr></table>",
+        "bbox": [
+            501,
+            599,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on Tiny ImageNet-C are shown in Table 5 and Table 6. From Table 5, we see that our method improves the mCE by $2 . 3 2 \\%$ compared with AugMix and $0 . 4 5 \\%$ when combined with DeepAugment. From Table 6, we can see our method outperforms ANT by a considerable margin. ",
+        "bbox": [
+            176,
+            702,
+            820,
+            743
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Training Time Since we use an accelerated adversarial training method [28] to solve Eq. (4), the worst-case mixing strategy can be searched efficiently. As a result, AugMax adds only a modest training overhead over AugMix, and is significantly more efficient than traditional adversarial training (AT). Detailed training time is shown in Table 7. ",
+        "bbox": [
+            174,
+            755,
+            578,
+            838
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/2ee30b11027501efa03bdd7216696bfcbd90117b011dd41d2c4257c7c17ce87a.jpg",
+        "table_caption": [
+            "Table 7: Training time on ImageNet with ResNet18, reported on a single NVIDIA A6000. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Time (sec/epoch)</td></tr><tr><td>Normal</td><td>2669</td></tr><tr><td>AugMix</td><td>3622</td></tr><tr><td>AT[6]</td><td>37162</td></tr><tr><td>AugMax</td><td>5264</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            856,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Analysis of Different Normalization Layers In this paragraph, we compare AugMax/AugMix when combined with different normalization schemes. When using DuBN or DuBIN, we route original images to the $B N _ { c }$ , and AugMax/AugMix images to $B N _ { a }$ . Since it is the by-default setting to use disentangled normalization layers (e.g., DuBN) for original and augmentation images in adversarial training [7, 64], we follow this traditional routing and combine AugMax with DuBN or DuBIN instead of BN or IBN. It is fair to compare AugMax-DuBN/DuBIN with either AugMixBN/IBN or AugMix-DuBN/DuBIN. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results are shown in Table 8. AugMax-DuBN outperforms both AugMix-BN and AugMixDuBN, and AugMax-DuBIN outperforms both AugMix-IBN and AugMix-DuBIN. Noticeably, when combined with AugMix, DuBIN also helps improve robustness. This shows the potential of applying DuBIN to other diversity augmentation methods for general performance improvement. ",
+        "bbox": [
+            174,
+            166,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also conduct stability analyses on these experiments, by showing the statistical significance of the improvements achieved by ",
+        "bbox": [
+            173,
+            297,
+            485,
+            340
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/134aa90dd51e15944e9ce97d82514165d2a9c3f133455c97cadde603c5616614.jpg",
+        "table_caption": [
+            "Table 8: Ablation results on DuBIN. RA $( \\% )$ on CIFAR10/100-C with WRN40-2 backbone are reported. Mean and standard derivation over three random seeds are shown for each experiment. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>CIFAR10-C</td><td>CIFAR100-C</td></tr><tr><td>AugMix-BN</td><td>89.01 (± 0.03)</td><td>64.56 (± 0.04)</td></tr><tr><td>AugMix-IBN</td><td>89.17 (± 0.24)</td><td>63.94 (± 0.28)</td></tr><tr><td>AugMix-DuBN</td><td>89.11 (± 0.21)</td><td>64.07 (± 0.38)</td></tr><tr><td>AugMix-DuBIN</td><td>89.74(± 0.35)</td><td>65.02 (± 0.58)</td></tr><tr><td>AugMax-DuBN</td><td>89.60 (± 0.62)</td><td>65.06 (± 0.28)</td></tr><tr><td>AugMax-DuBIN</td><td>90.67 (± 0.16)</td><td>66.35 (± 0.21)</td></tr></table>",
+        "bbox": [
+            500,
+            228,
+            821,
+            334
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "AugMax-DuBIN over the algorithm randomness. Specifically, we run all experiments in Table 8 using three different random seeds, and report the mean (denoted as $\\mu _ { . }$ ) and standard derivation (denoted as $\\sigma$ ) of accuracy in the form of $\\mu$ $( \\pm \\sigma )$ in Table 8. As we can observe, the improvements achieved by AugMax-DuBIN are statistically significant and consistent. ",
+        "bbox": [
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+            340,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "AugMax Hyperparameters In this paragraph, we check the sensitivity of AugMax with respect to its hyperparameters: maximization steps $n$ , step size $\\alpha$ and consistency loss tradeoff parameter $\\lambda$ . The results are shown in Table 9. We first fix $\\lambda = 1 0$ and try different values for $n$ and $k$ . We fix $k = 1$ when adjust",
+        "bbox": [
+            173,
+            417,
+            437,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/e98dd2bc541f7677c1502bb41917ed4f8ca77eea1683de8cdd714205a8498da7.jpg",
+        "table_caption": [
+            "Table 9: Ablation results on AugMax-DuBIN update step number $n$ and, early-stopping step $k$ and consistency loss tradeoff parameter $\\lambda$ . Results are reported on CIFAR10/CIFAR10-C with WRN40-2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">n</td><td colspan=\"3\"></td><td colspan=\"3\"></td></tr><tr><td></td><td>5</td><td>10</td><td>1</td><td>k2</td><td>3</td><td>1</td><td>入 10</td><td>15</td></tr><tr><td>SA(%)</td><td>95.78</td><td>95.68</td><td>95.68</td><td>95.60</td><td>95.51</td><td>95.90</td><td>95.68</td><td>95.88</td></tr><tr><td>RA (%)</td><td>90.49</td><td>90.67</td><td>90.67</td><td>90.38</td><td>90.21</td><td>88.89</td><td>90.67</td><td>90.30</td></tr></table>",
+        "bbox": [
+            452,
+            473,
+            821,
+            523
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ing $n$ and fix $n = 1 0$ when adjusting $k$ . As we can see, AugMax is stable with respect to different values of $n$ and $k$ . We use $n = 1 0$ and $k = 1$ as the default values since they empirically yield good results at low training overhead. We then fix $n = 1 0 , k = 1$ and adjust $\\lambda$ . We find robustness peaks at around $\\lambda = 1 0$ , which we set as the default value. ",
+        "bbox": [
+            174,
+            529,
+            825,
+            583
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Comparison with Different Diversity and Hardness Strategies In this paragraph, we study different strategies to combine diversity and hardness. The results confirm that diversity and hardness are two complementary dimensions and their proper unification boosts model robustness. ",
+        "bbox": [
+            174,
+            606,
+            825,
+            647
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we compare AugMax with baseline methods from each strategy group (i.e., diversity or hardness), and other possible strategies to combine both. Besides AugMix and Adversarial Training (e.g., PDGAT [6], FAT [28]), which are the two representative examples for diversity and hardness as shown in Figure 1, we also design the following baseline methods. (1) AugMix+PGDAT: using both AugMix and adversarial images for augmentation. This is a naive baseline to combine diversity with hardness. (2) ",
+        "bbox": [
+            174,
+            655,
+            485,
+            805
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/9cd94270f3007989b307ac992ef7405e97d86da3bf07dd37f2bb5a1478d552eb.jpg",
+        "table_caption": [
+            "Table 10: Results of different augmentation strategies on CIFAR100(-C) with ResNet18. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Strategy</td><td>SA(%)</td><td>RA(%)</td></tr><tr><td>Normal</td><td>-</td><td>77.99</td><td>48.46</td></tr><tr><td>AugMix [5]</td><td>(diversity)</td><td>78.23</td><td>62.67</td></tr><tr><td>PGDAT [6]</td><td rowspan=\"4\">(hardness)</td><td>60.94</td><td>47.71</td></tr><tr><td>FAT[28]</td><td>61.51</td><td>48.70</td></tr><tr><td>AdvMax</td><td>56.61</td><td>38.65</td></tr><tr><td>AugMix+PGDAT</td><td>61.68</td><td>51.39</td></tr><tr><td>AdvMix</td><td rowspan=\"2\">(diversity &amp;adversity)</td><td>72.36</td><td>56.89</td></tr><tr><td>AugMax-DuBIN</td><td>78.52</td><td>64.02</td></tr></table>",
+        "bbox": [
+            500,
+            686,
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+            803
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "AdvMix: applying adversarial attacks on the augmentation hyperparameters such as the rotation angles and translation pixel numbers [71, 72], while randomly selecting the mixing parameters; (3) AdvMax: applying adversarial attacks on both the augmentation hyperparameters and the mixing weights.5 Please see Appendix B for their implementation details. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The results are shown in Table 10. AugMax-DuBIN achieves the best performance, outperforming all methods from either diversity or hardness group. This shows diversity and hardness to be indeed complementary. On the other hand, the other two naive baselines jointly considering diversity and hardness achieve poorer performance, showing it nontrivial to design a method achieving good balance between diversity and hardness. ",
+        "bbox": [
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+            92,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.4 Robustness against Other Distribution Shifts ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            178,
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+            193
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Although AugMax mainly aims at improving model robustness against common corruptions, we find that it can also gain robustness against other types of distribution shifts. Specifically, we evaluate on the distribution shifts caused by the differences in data collection process using CIFAR10.1 [30], and against Spatial Transform adversarial Attacks (STA) [71] on CIFAR10-STA. We generate CIFAR10-STA by applying the worst-of- $k$ attack with $k = 1 0$ on CIFAR10 test set following [71]. ",
+        "bbox": [
+            174,
+            204,
+            531,
+            328
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/1a0663d0184d4b8bde087a73304f64306f952ffb350c3afa9da25e3f99babe16.jpg",
+        "table_caption": [
+            "Table 11: Robustness against other distribution shifts. Accuracy $( \\% )$ on CIFAR10.1 and CIFAR10-STA are evaluated on ResNeXt29 trained on CIFAR10. "
+        ],
+        "table_footnote": [
+            "Results are reported in Table 11 (using the same ResNeXt29 models trained on CIFAR10 as in Section 4.2). AugMax-DuBIN steadily outperforms AugMix against both distributional shifts. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>CIFAR10.1</td><td>CIFAR10-STA</td></tr><tr><td>Normal</td><td>88.90</td><td>30.30</td></tr><tr><td>AugMix</td><td>88.90</td><td>54.30</td></tr><tr><td>AugMax-DuBIN</td><td>90.64</td><td>63.20</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    {
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+        "text": "In this paper, we propose AugMax, a novel data augmentation strategy, that significantly improves robustness by strategically unifying diversity and hardness. To enable efficient training while facing the resulting heterogeneous features, we design a novel normalization scheme termed DuBIN. Using the combination of AugMax and DuBIN, we consistently demonstrate state-of-the-art robustness on several natural corruption benchmarks. ",
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+        "text": "AugMax unifies diversity and hardness in a heuristic manner. While we showed in ablation studies that AugMax outperforms other heuristic combinations, there may be better trade-offs available. We leave such study for future work and focus in this paper on showing the benefit of unifying diversity and hardness, which were separately considered in previous research. ",
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+        "type": "text",
+        "text": "Acknowledgement ",
+        "text_level": 1,
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+        "text": "Z.W. is supported by the U.S. Army Research Laboratory Cooperative Research Agreement W911NF17-2-0196 (IOBT REIGN), and an NVIDIA Applied Research Accelerator Program. ",
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+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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parse/train/S1x2PCNKDB/S1x2PCNKDB.md

@@ -0,0 +1,415 @@
+# TASK-RELEVANT ADVERSARIAL IMITATION LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We show that a critical problem in adversarial imitation from high-dimensional sensory data is the tendency of discriminator networks to distinguish agent and expert behaviour using task-irrelevant features beyond the control of the agent. We analyze this problem in detail and propose a solution as well as several baselines that outperform standard Generative Adversarial Imitation Learning (GAIL). Our proposed solution, Task-Relevant Adversarial Imitation Learning (TRAIL), uses a constrained optimization objective to overcome task-irrelevant features. Comprehensive experiments show that TRAIL can solve challenging manipulation tasks from pixels by imitating human operators, where other agents such as behaviour cloning (BC), standard GAIL, improved GAIL variants including our newly proposed baselines, and Deterministic Policy Gradients from Demonstrations (DPGfD) fail to find solutions, even when the other agents have access to task reward.
+
+# 1 INTRODUCTION
+
+Generative Adversarial Networks (GANs) have produced breath-taking conditional image synthesis results (Goodfellow et al., 2014; Brock et al., 2019), and have inspired adversarial learning approaches to imitating behavior. In Generative Adversarial Imitation Learning (GAIL) (Ho & Ermon, 2016), a discriminator network is trained to distinguish agent and expert behaviour through its observations, and is then used as a reward function. GAIL agents can overcome the exploration challenge by taking advantage of expert demonstrations, while also achieving high asymptotic performance by learning from agent experience.
+
+Despite the huge promise of GAIL, it has not yet had the same impact as GANs; in particular, robust GAIL from pixels for control applications remains a challenge. Here, we study a key shortcoming of GAIL: the tendency of the discriminator to mainly exploit task-irrelevant features. For example, by focusing on slight background differences, a discriminator can achieve perfect generalization, assigning zero reward to all held-out agent observations. However, this discriminator does not yield an informative reward function because it ignores behavior.
+
+![](images/21a1b85e2a28f46c1e51c7d95c791c9593dbf312cc715586c7be17a650a74728.jpg)  
+Figure 1: GAIL and TRAIL succeed at lifting (a), but when distractor objects are added, GAIL fails while TRAIL succeeds (b). Due to robustness to initial conditions, TRAIL can stack from pixels while standard GAIL fails (c). We witness this difference again in insertion with distractors (d). A video showing agents performing these tasks can be seen at https://youtu.be/Rz5G15rDKcg .
+
+![](images/862945eb66675f4de9e3964473bbe1038dacc00ee10aed40f2139ea575ebaafb.jpg)  
+Figure 2: Illustration of several task-irrelevant changes between the expert demonstrations and the distribution of agent observations, for the lift (red cube) task. The naively-trained discriminator network will use these differences rather than task performance to distinguish agent and expert.
+
+Assuming there is an expert policy $\pi _ { E }$ that is optimal for an unknown reward function, here we refer to a feature as task-irrelevant if it does not affect that reward. For example, if the task is to lift a red block, the positions of other blocks would be task-irrelevant; see Figures 1 and 2.
+
+This paper makes the following contributions:
+
+1. It reveals a fundamental limitation of GAIL by showing that discriminators do in practice exploit task-irrelevant information, thereby resulting in poor task performance.   
+2. It introduces powerful GAIL baselines. In particular, it shows that standard regularization and data augmentation are generally useful and improve upon standard GAIL.   
+3. It shows that these improvements to GAIL, as well as other improvements proposed by Reed et al. (2018), do not completely solve the problem, allowing GAIL agents to fail catastrophically with the addition of task-irrelevant distractors.   
+4. It introduces Task Relevant Adversarial Imitation Learning (TRAIL), using constrained optimization to force the discriminator to focus on the relevant aspects of the task, which improves performance dramatically on manipulation tasks from pixels (see Figure 1).
+
+# 2 RELATED WORK
+
+The use of demonstrations to help agent training has been studied extensively in robotics (Bakker & Kuniyoshi, 1996; Kawato et al., 1994; Miyamoto et al., 1996) with approaches ranging from Q-learning (Schaal, 1997) to behavioral cloning (BC) (Pomerleau, 1989).
+
+BC: BC is effective in solving many control problems (Pomerleau, 1989; Finn et al., 2017; Duan et al., 2017; Rahmatizadeh et al., 2018). It has also been successfully applied to initialize RL training (Rajeswaran et al., 2017). It, however, suffers from compounding errors as initial small deviations from the expert behaviors tend to cause bigger differences (Ross et al., 2011). This often necessitates a large number of demonstrations for satisfactory performance. Furthermore, BC typically does not lead to agents that are superior to their demonstrators.
+
+Inverse RL: Ziebart et al. (2008); $\mathrm { N g }$ et al. (2000); Abbeel & $\mathrm { N g } \left( 2 0 0 4 \right)$ propose inverse reinforcement learning (IRL) as a way of learning reward functions from demonstrations. Reinforcement learning can then be used to optimize that learned reward. Recently, Finn et al. (2016b) approached continuous robotic control problems with success by applying Maximum Entropy IRL algorithms which are very closely related to GAIL (Finn et al., 2016a) and have similar drawbacks.
+
+Learning from Demonstrations: Hester et al. (2018) developed deep Q-Learning from demonstration (DQfD), in which expert trajectories are added to experience replay and jointly used to train agents along with their own experiences. This was later extended by Vecerik et al. (2017) and Pohlen et al. (2018) to better handle sparse-reward problems in control and Atari games respectively. Despite their efficiency, this class of methods still requires access to rewards in order to learn.
+
+GAIL: Following the success of Generative Adversarial Networks (Goodfellow et al., 2014) in image generation, GAIL (Ho & Ermon, 2016) applies adversarial learning to the problem of imitation. Although many variants are introduced in the literature (Li et al., 2017; Fu et al., 2018; Merel et al., 2017; Zhu et al., 2018; Baram et al., 2017), making GAIL work for high-dimensional input spaces, particularly raw pixels, remains a challenge.
+
+A few papers (Peng et al., 2018; Reed et al., 2018; Blondé & Kalousis, 2018) seek to address the problem of overfitting the discriminator. Peng et al. (2018) introduces the Variational Bottleneck to regularize the discriminator. Reed et al. (2018) proposes to not train the vision module of the discriminator (e.g. use the vision module of the critic network instead) and only train a tiny network on top of the vision module to discriminate. Blondé & Kalousis (2018) follow a similar approach. Unstructured regularization, however, cannot stop the discriminator from fitting to features that are systematically different between the agents’ and the demonstration’s behavior, like those illustrated in Figure 2. TRAIL, on the other hand, is much less prone to overfitting to these features.
+
+Stadie et al. (2017) extend GAIL to the setting of third person imitation, in which the demonstrator and agent observations come from different views. To prevent the discriminator from discriminating based on viewpoint domain, they use gradient flipping from an auxiliary classifier to learn domain-invariant features. Our approach is not to learn domain-invariant features, but instead learn domain-agnostic discriminators that only focus on behavior.
+
+Several recent works have focused on improving the sample efficiency of GAIL (Blondé & Kalousis, 2018; Sasaki et al., 2018). Common to these approaches and to this work, is the use of off-policy actor critic agents and experience replay, to improve the utilization of available experience.
+
+# 3 REINFORCEMENT LEARNING AND ADVERSARIAL IMITATION
+
+Following the notation of Sutton & Barto (2018), a Markov Decision Process (MDP) is a tuple $( \mathcal { S } , \mathcal { A } , R , \mathbf { \bar { P } } , \gamma )$ with states $s$ , actions $\mathcal { A }$ , reward function $R ( s , a )$ , transition distribution $P ( s ^ { \prime } | s , a )$ , and discount $\gamma$ . An agent in state $s \in S$ takes action $a \in A$ according to its policy $\pi$ and moves to state $s ^ { \prime } \in \varDelta$ according to the transition distribution. The goal of RL algorithms is to find a policy that maximizes the expected sum of discounted rewards, represented by the action value function $\begin{array} { r } { Q ^ { \pi } ( s , a ) = \mathbb { E } ^ { \pi } [ \sum _ { t = 0 } ^ { \infty } \overset { \cdot } { \gamma } ^ { t } R ( s _ { t } , a _ { t } ) ] } \end{array}$ , where $\mathbb { E } ^ { \pi }$ is an expectation over trajectories starting from $s _ { 0 } = s$ and taking action $a _ { 0 } = a$ and thereafter running the policy $\pi$ .
+
+To apply RL, it is essential that we have access to the reward function which is often hard to design and evaluate (Singh et al., 2019). In addition, sparse rewards can cause exploration difficulties that pose great challenges to RL algorithms. We therefore look to imitation learning and particularly GAIL to derive a reward function from expert demonstrations. In GAIL, a reward function is learned by training a discriminator network $D ( s , a )$ to distinguish between agent and expert state-action pairs. The GAIL objective is thus formulated as follows:
+
+$$
+\operatorname* { m i n } _ { \pi } \operatorname* { m a x } _ { D } \mathbb { E } _ { ( s , a ) \sim \pi _ { E } } [ \log D ( s , a ) ] + \mathbb { E } _ { ( s , a ) \sim \pi } [ \log ( 1 - D ( s , a ) ) ] - \lambda _ { H } H ( \pi ) ,
+$$
+
+where $\pi$ is the agent policy, $\pi _ { E }$ the expert policy, and $H ( \pi )$ an (optional) entropy regularizer. The reward function is defined simply: $R ( s , a ) = - \log ( 1 - D ( s , a ) )$ .
+
+GAIL is theoretically appealing and practically simple. The discriminator, however, can focus on any features to discriminate, whether these features are task-relevant or not. In the next subsection we describe a way to constrain the discriminator network in order to prevent it from using task-irrelevant details to distinguish agent and expert data.
+
+# 4 TASK-RELEVANT ADVERSARIAL IMITATION LEARNING (TRAIL)
+
+We want the discriminator to focus on task-relevant features. Our proposed solution, TRAIL, prevents the discriminator from being able to distinguish expert and agent behaviour based on selected aspects of the data. For instance, the discriminator should distinguish agent and expert frames only when meaningful behavior is present in those frames. In the absence of behavior useful to solve the task, e.g. in initial frames prior to the execution of the behavior, the discriminator should be agnostic.
+
+To this end, we propose a constraint for the discriminator, such that its accuracy must not be greater than chance on a particular set of observations $\mathcal { L }$ that we will call the invariant set. $\mathcal { T }$ will include only observations that can be distinguished as agent or expert in task-irrelevant ways. Precisely, we formulate TRAIL in terms of the following constrained optimization problem:
+
+$$
+\begin{array} { r l } { \displaystyle \mathop { \operatorname* { m a x } } _ { \psi } } & { \mathbb { E } _ { s \sim \pi _ { E } } \left[ \log D _ { \psi } ( s ) \right] + \mathbb { E } _ { s \sim \pi _ { \theta } } \left[ \log ( 1 - D _ { \psi } ( s ) ) \right] } \\ { \displaystyle s . t . } & { \frac { 1 } { 2 } \mathbb { E } _ { s \sim \pi _ { E } } \left[ { \mathbf 1 } _ { { \mathbf D } _ { \psi } ( { \mathbf s } ) \geq \frac { 1 } { 2 } } \mid s \in { \mathbb Z } \right] + \frac { 1 } { 2 } \mathbb { E } _ { s \sim \pi _ { \theta } } \left[ { \mathbf 1 } _ { { \mathbf D } _ { \psi } ( { \mathbf s } ) < \frac { 1 } { 2 } } \mid s \in { \mathbb Z } \right] \le \frac { 1 } { 2 } . } \end{array}
+$$
+
+To apply the above constraint in practice, we can optimize the reverse of the objective function on $\mathcal { T }$ That is, given a batch of $N$ examples $s _ { e } \sim \pi _ { E } , s _ { \theta } \sim \pi _ { \theta }$ from the expert and agent, and $\hat { s } _ { e } \sim \pi _ { E }$ and $\hat { s } _ { \theta } \sim \pi _ { \theta }$ both in the set $\mathcal { T }$ , we maximize the following augmented objective function:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \psi } \big ( s _ { e } , s _ { \theta } , \hat { s } _ { e } , \hat { s } _ { \theta } \big ) = \sum _ { i = 1 } ^ { N } \log D _ { \psi } \left( s _ { e } ^ { ( i ) } \right) + \log \left( 1 - D _ { \psi } \left( s _ { \theta } ^ { ( i ) } \right) \right) } \\ & { \qquad - \lambda \left[ \sum _ { i = 1 } ^ { N } \log D _ { \psi } \left( \hat { s } _ { e } ^ { ( i ) } \right) + \log \left( 1 - D _ { \psi } \left( \hat { s } _ { \theta } ^ { ( i ) } \right) \right) \right] \mathbf { 1 } _ { a c c u r a c y ( \hat { s } _ { e } , \hat { s } _ { \theta } ) \geq \frac { 1 } { 2 } } , } \end{array}
+$$
+
+where $a c c u r a c y ( \cdot , \cdot )$ is defined as the average of discriminator accuracies:
+
+$$
+a c c u r a c y ( \hat { s } _ { e } , \hat { s } _ { \theta } ) = \frac { 1 } { 2 N } \sum _ { i = 1 } ^ { N } \left[ \mathbf { 1 } _ { \mathrm { D } _ { \psi } \left( \hat { \mathbf { s } } _ { e } ^ { ( \mathrm { i } ) } \right) \geq \frac { 1 } { 2 } } + \mathbf { 1 } _ { \mathrm { D } _ { \psi } \left( \hat { \mathbf { s } } _ { \theta } ^ { ( \mathrm { i } ) } \right) < \frac { 1 } { 2 } } \right] .
+$$
+
+The scalar $\lambda \geq 0$ is a tunable hyperparameter.
+
+We used single frames as states but sequences can also be used, and we do not require expert actions. Not requiring actions enables learning in very off-policy settings, where the action dimensions and distributions of the demonstrator (another robot or human) are different from those of the agent.
+
+# 4.1 THE SELECTION OF THE INVARIANT SET
+
+The selection of the invariant set $\mathcal { T }$ is a design choice. In general, we could always contrive nonstationary and adversarial ways of making this choice difficult. However, we argue that in many situations of great interest, including our robotic manipulation setup, it is easy to propose effective and very general invariant sets.
+
+A straightforward way to collect robot data is to execute a random policy. We can then use the resulting random episodes, for both expert and agent, to construct the invariant set $\mathcal { T }$ . Another way to construct $\mathcal { T }$ is to use early frames from both expert and agent episodes. Since in early frames little or no task behavior is apparent, this strategy turns out to be effective and no extra data has to be collected. This strategy also improves robustness with respect to variation in the initial conditions of the task; see for example block insertion in Figure 1(d).
+
+Importantly, if the set $\mathcal { T }$ captures some forms of irrelevance but not all forms, it will nonetheless always help in improving performance. In this regard, TRAIL will dominate its GAIL predecessor whenever the designer has some prior on what aspects of the data might be task irrelevant.
+
+# 5 EXPERIMENTS
+
+We focus on solving robot manipulation tasks. The environment implements two work-spaces: one with a Kinova Jaco arm (Jaco), and the other with a Sawyer arm (Sawyer). See supplementary material A.1 for a detailed description. Environment rewards, which are not used by GAIL-based methods, are sparse and equal to $+ 1$ for each step when a given task is solved and 0 otherwise. The maximum reward for the episode is 200, since this is the length of a single evaluation episode.
+
+Our agent is based on the off-policy D4PG algorithm (Barth-Maron et al., 2018) because of its stability and data-efficiency (see supplementary material A.6). Following Vecerik et al. (2017), we add expert demonstrations into the agents’ experience replay, and refer to the resulting RL algorithm as D4PG from Demonstrations (D4PGfD). For each task we collect 100 human demonstrations.
+
+Data augmentation Traditional data augmentation has proved beneficial in imitation learning (Berseth & Pal, 2019). Surprisingly, to the best of our knowledge, this has not been explicitly studied in prior publications on GAIL. However, we find that data augmentation is a generally useful component to prevent discriminator overfitting. It drastically improves the baseline GAIL agent, and is necessary to solve any of the harder manipulation tasks. We distort images by randomly changing brightness, contrast and saturation; random cropping and rotation; adding Gaussian noise. When multiple sensor inputs are available (e.g. multiple cameras), we also randomly drop out these inputs, but leaving at least one active.
+
+All discriminator-based methods in this work (including all baselines) use data augmentation except as indicated in the relevant ablations (see Section 5.2). We also considered regularizing the GAIL discriminator with spectral normalization (Miyato et al., 2018). It performed slightly better than GAIL, but still failed in the presence of distractor objects, and we thus omit spectral normalization in the main experiments for simplicity.
+
+Actor early stopping When the agent has learned the desired behavior, and the resulting data is used for training, the discriminator will become unable to distinguish expert and agent observations based only on behavior. This forces the discriminator to rely on task-irrelevant information.
+
+To avoid this scenario, we propose to restart each actor episode after a certain number of steps such that successful behavior is rarely represented in agent data. This enables the discriminator to recognize the goal condition, which appears frequently at the end of demonstration episodes, as representative of expert behavior. To avoid hand-tuning the stopping step number, we found that the discriminator score can be used to derive an adaptive stopping criterion. Concretely, we restart an episode if the discriminator score at the current step exceeds the median score of the episode so far for $T _ { p a t i e n c e }$ consecutive steps (in practice we set $T _ { p a t i e n c e } = 1 0$ ).
+
+We set invariant set hyperparameter $\lambda = 1$ and use adaptive early stopping for TRAIL. The ablation with $\lambda = 0$ , i.e. adaptive early stopping only, is referred to as TRAIL-0. Hence, the only difference between TRAIL-0 and TRAIL is that the later uses invariant set constraints.
+
+# 5.1 BLOCK LIFTING WITH DISTRACTORS
+
+In this section, we consider two variants of the lift task in the Sawyer work space: a) lift alone, where only one red cube is present, b) lift distracted, with two extra blocks (blue and green, see Figure 9). We show how adding these additional distractors affects the training procedure.
+
+We first compare our method to baselines. In doing so, the invariant set is constructed using the first 10 frames from every episode, as in the rest of the work. This choice does not require us to collect any extra data, and hence the comparison with baselines is fair. In the next section, we elaborate on the choice of the invariant set and provide additional experimental results.
+
+As baselines, we run BC and GAIL (with data augmentation). Baseline GAIL is the strongest we were able to implement. The only difference between this baseline and TRAIL is the use of invariant set constrains and actor early stopping (code, agent configurations like number of actors, and network architecture are the same). We additionally consider the approaches proposed by Reed et al. (2018) as GAIL-based baselines; using either a randomly initialized convolutional network, or a convolutional critic network, to provide fixed vision features on top of which a tiny discriminator network is trained. We call these two baselines random and critic respectively. Finally, to show the importance of actor early stopping, we run TRAIL-0 (Fig. 3).
+
+![](images/59bb3bff9b0dfb40d7b0e4e3cc9fbc548f0f04232c09b94486607c7ff7cfd013.jpg)  
+Figure 3: Results for lift alone, lift distracted, and lift distracted seeded. Only TRAIL excels.
+
+All methods perform satisfactorily on lift alone, but the proposed methods TRAIL-0 and TRAIL do best. As expected, the performance of BC on lift distracted is similar to its performance on lift alone, despite the two additional blocks. The two additional blocks in lift distracted affect the GAIL-based baselines, despite being irrelevant to the task.
+
+To understand this effect, we conducted an additional experiment (lift distracted seeded). Here the initial block positions are randomly drawn from the expert demonstrations. Therefore, it is impossible to discriminate between expert and actor episodes using the first few frames of an episode. Note this initialization procedure is not applied to the evaluation actor, keeping the evaluation scores comparable between lift distracted and lift distracted seeded.
+
+This experiment exposes one major culprit behind the performance degradation of GAIL: memorization. The discriminator can achieve perfect accuracy by memorizing all 100 initial positions from the demonstration set, making the reward function uninformative. By constraining the discriminator, TRAIL squeezes out this irrelevant information and succeeds in solving the task in the presence of distractions. TRAIL is the only method that is able to handle the variety of initial cube positions during training, achieving better than expert performance on lift distracted.
+
+Interestingly, random performs reasonably on lift distracted. Given random’s strong performance, we conducted additional experiments to evaluate its effectiveness when trained with adaptive early stopping and present the results in Figure 12 of the supplementary material.
+
+# CONSTRUCTING THE INVARIANT SET $\mathcal { T }$
+
+In the previous subsection, early frames were used to construct the invariant set (TRAIL-early). Here, we evaluate another previously mentioned approach for constructing $\mathcal { T }$ ; random policy (TRAILrandom). The lift distracted task caused all baselines to fail, but was solved by TRAIL. We introduce a harder version of the task, where the expert appearance is different, to tease out the differences between TRAIL-early and TRAIL-random. The difference in appearance between the expert and imitator allows the GAIL discriminator to trivially distinguish them. The results and the differences in the expert appearance are presented in Figure 4.
+
+![](images/6b8cf409f166f495f4aaf1c532472a37e5e94c105788ca3abc316459da5a2a6d.jpg)  
+Figure 4: Lift red block, where expert has a different body appearance, and with distractor blocks. TRAIL-random outperforms GAIL, and performs on par with TRAIL-early.
+
+The new task is indeed harder and it takes longer for TRAIL methods to achieve performance better than BC baseline, which is not affected by the different body appearance. GAIL is clearly outperformed and does not take off.
+
+The difference between TRAIL methods is negligible. We also tried to mix them but the differences remain imperceptible. Hence, in the following experiments we simply use early frames. This choice is pragmatic as it does not require that we collect any extra data, and hence the comparison with GAIL and other baselines is fair. It is also very simple to apply in practice, even if one does not have access to the expert setup anymore. Finally, it is general and powerful enough to be successfully used across all robotic manipulation tasks considered in this work.
+
+To decide how many initial frames should be used to construct $\mathcal { T }$ , we conducted an ablation study and found out that the method is not very sensitive to this choice (see supplementary material A.2). Hence, we chose 10 initial frames, and intentionally used the same number for all tasks to further emphasize generality of this choice.
+
+# 5.2 ABLATION STUDIES
+
+# MEASURING DISCRIMINATOR MEMORIZATION OF TASK-IRRELEVANT FEATURES
+
+In this section, we experimentally confirm that memorization is a limiting factor of GAIL methods. We equipped the discriminator with two extra heads whose inputs are the final spatial layer of the
+
+ResNet for the lift distracted task. The first head is trained on the first frames only, and has the same target as the main head (i.e. discriminating between agent and expert). To train the second head, we randomly divide the expert demonstrations into two equi-numerous subsets, and the task is to predict to which of these randomly chosen sets the demonstration was assigned to. Both heads are trained via backpropagation but their gradient is not propagated to the ResNet so they do not influence the training procedure directly.
+
+If our claim is correct, we expect the extra heads to have higher accuracy for TRAIL-0 as compared to TRAIL, since TRAIL representations are penalized for having features triggering memorization, i.e. the features should not aid discriminating based on the first frames or predicting a random label for each expert demonstration. No reasonably performing method is able to force $50 \%$ accuracy for extra heads since some features are important to solve the task (e.g. the position of the red cube).
+
+We also collected 25 extra holdout demonstrations and visualize the average discriminator prediction on them, and compare with predictions on the training demonstrations.
+
+![](images/f3d0ef0720ecd274b57537e078d92668d28a832dccb1c8bad24328a24d337bc8.jpg)  
+Figure 5: Demonstrating the memorization problem on the lift distracted task (here higher accuracy is worse). Accuracy of different discriminator heads is presented (A-D). In A, the overall accuracy for all timesteps. Then main $\mathrm { ( m ) }$ and extra (e) heads accuracy for the first steps are presented in B and C, respectively. Accuracy of the head predicting randomly assigned demonstration class is shown in D. Average discriminator predictions for training and holdout demonstration are shown in E and F.
+
+In Fig. 5, we see the overall accuracy of the main head for TRAIL-0 is significantly higher, and the difference is larger for early steps (when TRAIL achieves only $50 \%$ , as expected). TRAIL representations are less helpful for extra heads (see Figure 5 C-D). Finally, TRAIL-0 clearly overfits on training demonstrations, predicting almost the maximum score, while only $\sim 0 . 2 5$ is predicted for holdout demonstrations. The TRAIL average predictions for both datasets are almost identical.
+
+ACTOR EARLY STOPPING (TRAIL-0)
+
+In this section, we analyze the importance of adaptive early stopping on 3 tasks in the Jaco workspace: lift red cube $( l i f t )$ , put red cube in box $( b o x )$ , and stack red cube on blue cube (stack). We consider D4PGfD and three GAIL-based models with varying termination policies: a) fixed step (50), b) based on ground truth task rewards, and c) based on adaptive early stopping (TRAIL-0). Using ground truth task rewards, an episode is terminated if the reward at the current step exceeds the median reward of the episode so far for 10 consecutive steps. Results are presented in Figure 6.
+
+Termination based on task reward is clearly superior; although unrealistic in practice, it defines the performance upper-bound and clearly shows that early stopping is beneficial. TRAIL-0 is robust and reaches human performance on all tasks. A fixed termination policy, when tuned, can be very effective. The same fixed termination step, however, does not work for all tasks. See Figure 13 in supplementary material for the effects of varying termination steps. Finally, as can be inferred from stack results, the dense rewards provided by TRAIL-0 are helpful in solving this challenging problem which is unsolved with D4PGfD even though D4PGfD uses ground truth task rewards.
+
+![](images/236f966ae10dec709f176a7097a72d5e3722052c12c4e9e616a6e0ea34d38c6a.jpg)  
+Figure 6: Results for lift, box, and stack on Jaco environment.
+
+# DATA AUGMENTATION
+
+In Table 5.2, we report the best reward obtained in the first 12 hours of training (averaged for all seeds;   
+see Figure 14 for full curves). The results show that data augmentation is needed in lift distracted.   
+For the lift alone task, TRAIL-0 with data augmentation performs on par with TRAIL.
+
+Table 1: Influence of data augmentation (evaluated on rewards) for lift alone and lift distracted.   
+
+<table><tr><td>Task</td><td>Regularization</td><td>Data augmentation</td><td>No data augmentation</td></tr><tr><td>lift alone</td><td>TRAIL-0 TRAIL</td><td>~165</td><td>~115</td></tr><tr><td rowspan="2">lift distracted</td><td>TRAIL-0</td><td>~155</td><td>~165</td></tr><tr><td>TRAIL</td><td>~30 ~180</td><td>~5 ~10</td></tr></table>
+
+The performance of TRAIL is not affected by the lack of data augmentation on the lift alone task, whereas the performance of TRAIL-0 is.
+
+# LEARNING WITH A FIXED, PERFECT DISCRIMINATOR
+
+To assess whether learned discriminators are necessary, we compare TRAIL against agents using a fixed reward function corresponding to $R _ { e x p e r t } = 1$ and $R _ { a g e n t } = 0$ for the lift alone and lift distracted tasks. This baseline simulates an oracle discriminator with perfect generalization, but which is agnostic to behavior. On the lift alone task, agents using this fixed reward achieve roughly half the reward of TRAIL asymptotically, and on lift distracted they do not solve the task (average rewards are less than 5). See supplementary Figure 16 for learning curves.
+
+# 5.3 LEARNING FROM OTHER EMBODIMENTS AND PROPS
+
+Since the TRAIL discriminator is trained to ignore task-irrelevant features, it can learn from demonstrations with different embodiments and props. Figure 7 shows that GAIL even with augmentation fails to learn block lifting from a different embodiment, and performs worse when the expert uses a different prop color. TRAIL solves the task and achieves better performance in both cases.
+
+![](images/c312fba1da06c95e98c0f3e62ac762e79d2ef8e05c275905eb6accf57d14aeb2.jpg)  
+Figure 7: When the expert differs in body or prop appearance, TRAIL outperforms GAIL.
+
+# 5.4 EVALUATION ON DIVERSE MANIPULATION TASKS
+
+To further demonstrate benefits of using our proposed method, we present results for TRAIL, TRAIL0, the baseline GAIL, and BC on more challenging tasks. Specifically, we consider stack with the Sawyer robot; and insertion and stack banana in the Jaco work-space. The results are shown in Figure 8. The tasks we consider here are much harder as evidenced by the performance of BC agents. These experiments suggest that TRAIL is generally useful as an improvement over GAIL, even when the tasks are not designed to include task-irrelevant information.
+
+![](images/b34a66b68ab4b752173566a72eef0ab16bcf79a532090242dd93629c664709c9.jpg)  
+Figure 8: Results comparing TRAIL, TRAIL-0 and GAIL for diverse manipulation tasks.
+
+# 6 CONCLUSIONS
+
+To make adversarial imitation work on nontrivial tasks from pixels, it is crucial to prevent the discriminator from exploiting task-irrelevant information. Data augmentation is a necessary component that prevents the discriminator from overfitting, but alone is not sufficient to solve more challenging problems. Our proposed method TRAIL proved to be a remedy which effectively focuses the discriminator on the task even when task-irrelevant features are present.
+
+We showed through ablations that TRAIL benefits from data augmentation, actor early stopping, and the use of invariant set constraints. The last enables our agent to solve the full task suite in our experiments showing its decisive importance.
+
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+
+# A SUPPLEMENTARY MATERIAL
+
+# A.1 DETAILED DESCRIPTION OF ENVIRONMENT
+
+All our simulations are conducted using $\mathbf { M u J o C o ^ { 1 } }$ (Todorov et al., 2012). We test our proposed algorithms in a variety of different envrionments using simulated Kinova Jaco2, and Sawyer robot arms3; see Figure 9. We use the Robotiq 2F85 gripper4 in conjuction with the Sawyer arm.
+
+![](images/6bccf9982d118787a6c2979868292057dd3b8d908253da25a031063cb76c20dc.jpg)
+
+Figure 9: Two work spaces, Jaco (left) which uses the Jaco arm and is $2 0 \times 2 0 ~ \mathrm { c m }$ , and Sawyer (right) which uses the Sawyer arm and more closely resembles a real robot cage and is $3 5 \times 3 5 \mathrm { c m }$ .
+
+To provide demonstrations, we use the SpaceNavigator 3D motion controller5 to set Cartesion velocities of the robot arm. The gripper actions are implemented via the buttons on the controller. All demonstrations in our experiments are provided via human teleoperation and we collected 100 demonstrations for each experiment.
+
+Jaco: When using the Jaco arm, we use joint velocity control (9DOF) where we control all 6 joints of arm and all 3 joints of the hand. The simulation is run with a numerical time step of 10 milliseconds, integrating 5 steps, to get a control frequency of 20HZ. The agent uses a frontal camera of size $6 4 \times 6 4$ (see Figure 10(a)). For a full list of observations the agent sees, please refer to Table 2(a).
+
+![](images/b6bc13e4b98119453319a74daf7dde365754b163a2a6e1e98dd817c38debb41d.jpg)  
+Figure 10: Illustration of the pixels inputs to the agent.
+
+Sawyer: When using the Sawyer arm, we use Cartesian velocity control (6DOF) for the robot arm and add one additional action for the gripper resulting in 7 degrees of freedom. The simulation is run with a numerical time step of 10 milliseconds, integrating 10 steps, to get a control frequency of 10HZ. The agent uses two frontal cameras of size $6 4 \times 6 4$ situated on the left and right side of the robot cage respectively (see Figure $1 0 ( \mathrm { b } , \mathrm { c } ) ,$ ). For a full list of observations the agent sees, please refer to Table 2(b).
+
+For all environments considered in this paper, we provide sparse rewards (i.e. if task is accomplished, the reward is 1 and 0 otherwise). In experiments regards our proposed methods, rewards are only used for evaluation purposes and not for training the agent.
+
+a) Jaco   
+
+<table><tr><td>Feature Name</td><td>Dimensions</td></tr><tr><td>frontal camera base force and torque sensors</td><td>64×64×3 6</td></tr><tr><td>arm joints position</td><td>6</td></tr><tr><td>arm joints velocity</td><td>6</td></tr><tr><td>wrist force and torque sensors</td><td>6</td></tr><tr><td>hand finger joints position</td><td>3</td></tr><tr><td>hand finger joints velocity</td><td></td></tr><tr><td>hand fingertip sensors</td><td>3</td></tr><tr><td></td><td>3</td></tr><tr><td>grip site position</td><td>3</td></tr><tr><td>pinch site position</td><td>3</td></tr></table>
+
+Table 2: Observation and dimensions.   
+
+<table><tr><td colspan="2">b) Sawyer</td></tr><tr><td>Feature Name frontleft camera</td><td>Dimensions 64×64×3</td></tr><tr><td>front right camera</td><td>64×64×3</td></tr><tr><td>arm joint position</td><td>7</td></tr><tr><td>arm joint velocity</td><td>7</td></tr><tr><td>wrist force sensor</td><td>3</td></tr><tr><td>wrist torque sensor</td><td>3</td></tr><tr><td>hand grasp sensor</td><td></td></tr><tr><td></td><td>1</td></tr><tr><td>hand joint position</td><td>1</td></tr><tr><td>tool center point cartesian orientation</td><td>9</td></tr><tr><td>tool center point cartesian position</td><td></td></tr><tr><td>hand joint velocity</td><td>3 1</td></tr></table>
+
+# A.2 EARLY FRAMES ABLATION STUDY
+
+Here, we vary the number of early frames of each episode used to form $\mathcal { T }$ . We report that a range of values from 1 up to 20 works well across both tasks (Put in box and Stack banana), with performance gradually degrading as the value increased beyond 20 (Fig. 11).
+
+![](images/62c162c76431293f01240e50adc3fc2894bfed770d528775da6f48cfa37dd471.jpg)  
+Figure 11: TRAIL performance, varying the number of first frames in each episode used to form $\mathcal { T }$ .
+
+# A.3 TRAIL-0 WITH random
+
+We found out and mentioned in the subsection 5.1 that random benefits from TRAIL-0. Unfortunately, it is still prone to overfitting and hence, worse than our full method – TRAIL. We present random $^ +$ TRAIL-0 accompanied with our methods in Figure 12. Our TRAIL and random $^ +$ TRAIL-0 are the only methods exceeding BC performance on lift distracted. However, TRAIL performance is clearly better (obtains higher rewards and never gets overfitted).
+
+![](images/ed344e1142f29ef99b9a64add4b50040c0bf4ab9992f0071804687203c06a46a.jpg)  
+Figure 12: Results for lift alone, lift distracted, and lift distracted seeded.
+
+# A.4 FIXED TERMINATION POLICY
+
+As mentioned in the subsection 5.2, the most basic early termination policy – fixed step termination – may be very effective if tuned. Since the tuning may be expensive in practice, we recommend using adaptive early stopping (TRAIL-0). However, for the sake of completeness we provide results for fixed step termination policy depending on the hyperparameter tuned. The results for stack task are presented in Figure 13. The Jaco work space is considered here because Sawyer requires TRAIL to obtain high rewards. As can be inferred from the figure, the performance is very sensitive to the fixed step hyperparameter. We refer to subsection 5.2 for more comments on all methods.
+
+![](images/6ce8e8947b35ee83aff9d74c4c2c443c36dcd3ab4361b7b7f06e4556241f9585.jpg)  
+Figure 13: Results for stack in Jaco work space. Fixed step termination policy can be very effective but the final performance is very sensitive to the hyperparameter. TRAIL-0 does not need tuning nor access to the environment reward.
+
+# A.5 DATA AUGMENTATION
+
+An extra set of experiments on lift alone and lift distracted tasks (described in subsection 5.1) has been performed to show importance of data augmentation.
+
+Because in the subsection 5.2 only the peak performance is presented (Table 5.2), we present here the full curves in Figure 14. The results shows that data augmentation is necessary to obtain high rewards in lift distracted. For easier lift alone, TRAIL-0 with data augmentation perform at par with TRAIL.
+
+![](images/bb24dedec05c204244af33e7fd3602ca1d47b8b427d2224cef1cd3839e7e72d0.jpg)  
+Figure 14: Results for lift alone and lift distracted in Sawyer work space. TRAIL and TRAIL-0 are by default with data augmentation. Additional results for these methods without data augmentation are presented to show its importance.
+
+# A.6 D4PG
+
+We use D4PG (Barth-Maron et al., 2018) as our main training algorithm. Briefly, D4PG is a distributed off-policy reinforcement learning algorithm for continuous control problems. In a nutshell, D4PG uses Q-learning for policy evaluation and Deterministic Policy Gradients (DPG) (Silver et al., 2014) for policy optimization. An important characteristic of D4PG is that it maintains a replay memory $\mathcal { M }$ (possibility prioritized (Horgan et al., 2018)) that stores SARS tuples which allows for off-policy learning. D4PG also adopts target networks for increased training stability. In addition to these principles, D4PG utilized distributed training, distributional value functions, and multi-step returns to further increase efficiency and stability. In this section, we explain the different ingredients of D4PG.
+
+D4PG maintains an online value network $Q ( s , a | \theta )$ and an online policy network $\pi ( s | \phi )$ . The target networks are of the same structures as the value and policy network, but are parameterized by different parameters $\theta ^ { \prime }$ and $\phi ^ { \prime }$ which are periodically updated to the current parameters of the online networks.
+
+Given the $Q$ function, we can update the policy using DPG:
+
+$$
+\mathcal { I } ( \phi ) = \mathbb { E } _ { s _ { t } \sim \mathcal { M } } \left[ \nabla _ { \phi } Q ( s _ { t } , \pi ( s _ { t } | \phi ) | \theta ) \right] .
+$$
+
+Instead of using a scalar $Q$ function, D4PG adopts a distributional value function such that $Q ( s _ { t } , a | \theta ) ~ = ~ \bar { \mathbb { E } } \big [ Z ( s _ { t } , a | \theta ) \big ]$ where $Z$ is a random variable such that $Z ~ = ~ z _ { i }$ w.p. $\qquad p _ { i } \propto$ $\exp ( \omega ( s _ { t } , a | \theta ) )$ . The $z _ { i }$ ’s take on $V _ { b i n s }$ discrete values that ranges uniformly between $V _ { m i n }$ and $V _ { m a x }$ such that $\begin{array} { r } { z _ { i } = V _ { m i n } + i \frac { V _ { m a x } - V _ { m i n } } { V _ { b i n s } } } \end{array}$ for $i \in \{ 0 , \cdots , V _ { b i n s } - 1 \}$ .
+
+To construct a bootstrap target, D4PG uses $\mathbf { N } .$ -step returns. Given a sampled tuple from the replay memory: $s _ { t } , a _ { t } , \{ r _ { t } , r _ { t + 1 } , \cdot \cdot \cdot , r _ { t + N - 1 } \} , s _ { t + N }$ , we construct a new random variable $Z ^ { \prime }$ such that Z0 = zi + PN−1n=0 w.p. $p _ { i } \propto \exp ( \omega ( s _ { t + N } , \pi ( s _ { t + N } | \phi ^ { \prime } ) | \theta ^ { \prime } ) )$ . Notice, $Z ^ { \prime }$ no longer has the same support. We therefore adopt the same projection $\Phi$ employed by Bellemare et al. (2017). The training loss for the value function
+
+$$
+\mathcal { L } ( \theta ) = \mathbb { E } _ { s _ { t } , a _ { t } , \{ r _ { t } , \cdots , r _ { t + N - 1 } \} , s _ { t + N } \sim \mathcal { M } } \big [ H \big ( \Phi ( Z ^ { \prime } ) , Z ( s _ { t } , a _ { t } | \theta ) \big ) \big ] ,
+$$
+
+where $H$ is the cross entropy.
+
+D4PG is also distributed following Horgan et al. (2018). Since all learning processes only rely on the replay memory, we can easily decouple the ‘actors’ from the ‘learners’. D4PG therefore uses a large number of independent actor processes which act in the environment and write data to a central replay memory process. The learners could then draw samples from the replay memory for learning. The learner also serves as a parameter server to the actors which periodically update their policy parameters from the learner.
+
+In our experiments, we always have access to expert demonstrations. We, therefore adopt the practice from DQfD and DDPGfD and put the demonstrations into our replay buffers. For more details see Algorithms 1, 2.
+
+# Algorithm 1 Actor
+
+Given: • an experience replay memory $\mathcal { M }$   
+for $n _ { e p i s o d e s }$ do for $t = 1$ to $T$ do Sample action from task policy: $\boldsymbol { a } _ { t } \gets \pi ( \boldsymbol { s } _ { t } )$ Execute action $a _ { t }$ and observe new state $s _ { t + 1 }$ , and reward $r _ { t }$ Store transition $( s _ { t } , a _ { t } , r _ { t } , s _ { t + 1 } )$ in memory $\mathcal { M }$ end for   
+end for
+
+# Algorithm 2 Learner
+
+# Given:
+
+• an off-policy RL algorithm A • a replay buffer $\mathcal { M }$ • a replay buffer of expert demonstrations $\mathcal { M } _ { e }$
+
+Initialize A   
+for $n _ { u p d a t e s }$ do Sample transitions $\left( { { s _ { t } } , { a _ { t } } , { r _ { t } } , { s _ { t + 1 } } } \right)$ from $\mathcal { M }$ to make a minibatch $B$ . Sample transitions $\left( { { s _ { t } } , { a _ { t } } , { r _ { t } } , { s _ { t + 1 } } } \right)$ from $\mathcal { M } _ { e }$ enlarge the minibatch $B$ . Perform a actor update step with Eqn. (5). Perform a critic update step with Eqn. (6). Update the target actor/critic networks every $k$ steps.   
+end for
+
+# A.7 NETWORK ARCHITECTURE AND HYPERPARAMETERS
+
+Actor and critic share a residual pixel encoder network with eight convolutional layers (3x3 convolutions, three 2-layer blocks with 16, 32, 32 channels), instance normalization (Ulyanov et al., 2016) and exponential linear units (Clevert et al., 2015) between layers.
+
+The policy is a 3-layer MLP with ReLU activations with hidden layer sizes (300, 200). The critic is a 3-layer MLP with ReLU activations with hidden layer sizes (400, 300). For a illustration of the network. Please see Figure 15.
+
+![](images/28c5f55c9a59fb03ff40780ca1b034a37b8cd5bc60d7b421bfe54857e6457a48.jpg)  
+Figure 15: Network architecture for the policy and critic.
+
+The discriminator network uses a pixel encoder of the same architecture as the actor critic, followed by a 3-layer MLP with ReLU activations and hidden layer sizes (32, 32).
+
+Table 3: Hyper parameters used in robot manipulation experiments.   
+
+<table><tr><td>Parameters</td><td>Values</td></tr><tr><td>Actor/Critic Input Width</td><td>64</td></tr><tr><td>Actor/Critic Input Height</td><td>64</td></tr><tr><td>D4PG Parameters</td><td></td></tr><tr><td>Vmin</td><td>-50</td></tr><tr><td>Vmax</td><td>150</td></tr><tr><td>Vbins</td><td>21</td></tr><tr><td>N step</td><td>1</td></tr><tr><td>Actor learning rate</td><td>10-4</td></tr><tr><td>Critic learning rate</td><td>10-4</td></tr><tr><td>Optimizer</td><td>Adam (Kingma &amp; Ba (2014))</td></tr><tr><td>Batch size</td><td>256</td></tr><tr><td>Target update period</td><td>100</td></tr><tr><td>Discount factor (y)</td><td>0.99</td></tr><tr><td>Replay capacity</td><td>106</td></tr><tr><td>Number of actors</td><td>32 or 128</td></tr><tr><td>Imitation Parameters</td><td></td></tr><tr><td>Discriminator learning rate</td><td>10-4</td></tr><tr><td>Discriminator Input Width</td><td>48</td></tr><tr><td>Discriminator Input Height</td><td>48</td></tr></table>
+
+# A.8 COMPARING AGAINST LEARNING WITH FIXED REWARDS
+
+To check whether learned discriminators are needed, we compare TRAIL against agents using a fixed reward function corresponding to $R _ { e x p e r t } = 1$ and $R _ { a g e n t } = 0$ for the lift alone and lift distracted tasks. This baseline simulates an oracle discriminator with perfect generalization, but which is agnostic to behavior. On the lift alone task, agents using this fixed reward achieve roughly half the reward of TRAIL asymptotically, and on lift distracted they do not solve the task (average rewards are less than 5). The learning curves are presented in Figure 16.
+
+![](images/73412ac4784215fb9164009bf367fa91028931791c266c531f38108dcd112e50.jpg)  
+Figure 16: With fixed rewards, the agent is able to learn lift alone somewhat, but performs worse than TRAIL. When distractor blocks are added, the fixed reward agent fails to learn completely.

parse/train/S1x2PCNKDB/S1x2PCNKDB_content_list.json

@@ -0,0 +1,2216 @@
+[
+    {
+        "type": "text",
+        "text": "TASK-RELEVANT ADVERSARIAL IMITATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            99,
+            834,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            209,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We show that a critical problem in adversarial imitation from high-dimensional sensory data is the tendency of discriminator networks to distinguish agent and expert behaviour using task-irrelevant features beyond the control of the agent. We analyze this problem in detail and propose a solution as well as several baselines that outperform standard Generative Adversarial Imitation Learning (GAIL). Our proposed solution, Task-Relevant Adversarial Imitation Learning (TRAIL), uses a constrained optimization objective to overcome task-irrelevant features. Comprehensive experiments show that TRAIL can solve challenging manipulation tasks from pixels by imitating human operators, where other agents such as behaviour cloning (BC), standard GAIL, improved GAIL variants including our newly proposed baselines, and Deterministic Policy Gradients from Demonstrations (DPGfD) fail to find solutions, even when the other agents have access to task reward. ",
+        "bbox": [
+            233,
+            247,
+            766,
+            414
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            455,
+            334,
+            472
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative Adversarial Networks (GANs) have produced breath-taking conditional image synthesis results (Goodfellow et al., 2014; Brock et al., 2019), and have inspired adversarial learning approaches to imitating behavior. In Generative Adversarial Imitation Learning (GAIL) (Ho & Ermon, 2016), a discriminator network is trained to distinguish agent and expert behaviour through its observations, and is then used as a reward function. GAIL agents can overcome the exploration challenge by taking advantage of expert demonstrations, while also achieving high asymptotic performance by learning from agent experience. ",
+        "bbox": [
+            174,
+            492,
+            825,
+            590
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Despite the huge promise of GAIL, it has not yet had the same impact as GANs; in particular, robust GAIL from pixels for control applications remains a challenge. Here, we study a key shortcoming of GAIL: the tendency of the discriminator to mainly exploit task-irrelevant features. For example, by focusing on slight background differences, a discriminator can achieve perfect generalization, assigning zero reward to all held-out agent observations. However, this discriminator does not yield an informative reward function because it ignores behavior. ",
+        "bbox": [
+            173,
+            597,
+            826,
+            680
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/21a1b85e2a28f46c1e51c7d95c791c9593dbf312cc715586c7be17a650a74728.jpg",
+        "image_caption": [
+            "Figure 1: GAIL and TRAIL succeed at lifting (a), but when distractor objects are added, GAIL fails while TRAIL succeeds (b). Due to robustness to initial conditions, TRAIL can stack from pixels while standard GAIL fails (c). We witness this difference again in insertion with distractors (d). A video showing agents performing these tasks can be seen at https://youtu.be/Rz5G15rDKcg . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            705,
+            815,
+            857
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/862945eb66675f4de9e3964473bbe1038dacc00ee10aed40f2139ea575ebaafb.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of several task-irrelevant changes between the expert demonstrations and the distribution of agent observations, for the lift (red cube) task. The naively-trained discriminator network will use these differences rather than task performance to distinguish agent and expert. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            82,
+            789,
+            213
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Assuming there is an expert policy $\\pi _ { E }$ that is optimal for an unknown reward function, here we refer to a feature as task-irrelevant if it does not affect that reward. For example, if the task is to lift a red block, the positions of other blocks would be task-irrelevant; see Figures 1 and 2. ",
+        "bbox": [
+            176,
+            284,
+            825,
+            325
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper makes the following contributions: ",
+        "bbox": [
+            174,
+            333,
+            473,
+            347
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. It reveals a fundamental limitation of GAIL by showing that discriminators do in practice exploit task-irrelevant information, thereby resulting in poor task performance.   \n2. It introduces powerful GAIL baselines. In particular, it shows that standard regularization and data augmentation are generally useful and improve upon standard GAIL.   \n3. It shows that these improvements to GAIL, as well as other improvements proposed by Reed et al. (2018), do not completely solve the problem, allowing GAIL agents to fail catastrophically with the addition of task-irrelevant distractors.   \n4. It introduces Task Relevant Adversarial Imitation Learning (TRAIL), using constrained optimization to force the discriminator to focus on the relevant aspects of the task, which improves performance dramatically on manipulation tasks from pixels (see Figure 1). ",
+        "bbox": [
+            210,
+            358,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            530,
+            341,
+            545
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The use of demonstrations to help agent training has been studied extensively in robotics (Bakker & Kuniyoshi, 1996; Kawato et al., 1994; Miyamoto et al., 1996) with approaches ranging from Q-learning (Schaal, 1997) to behavioral cloning (BC) (Pomerleau, 1989). ",
+        "bbox": [
+            174,
+            560,
+            825,
+            603
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "BC: BC is effective in solving many control problems (Pomerleau, 1989; Finn et al., 2017; Duan et al., 2017; Rahmatizadeh et al., 2018). It has also been successfully applied to initialize RL training (Rajeswaran et al., 2017). It, however, suffers from compounding errors as initial small deviations from the expert behaviors tend to cause bigger differences (Ross et al., 2011). This often necessitates a large number of demonstrations for satisfactory performance. Furthermore, BC typically does not lead to agents that are superior to their demonstrators. ",
+        "bbox": [
+            173,
+            609,
+            825,
+            693
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Inverse RL: Ziebart et al. (2008); $\\mathrm { N g }$ et al. (2000); Abbeel & $\\mathrm { N g } \\left( 2 0 0 4 \\right)$ propose inverse reinforcement learning (IRL) as a way of learning reward functions from demonstrations. Reinforcement learning can then be used to optimize that learned reward. Recently, Finn et al. (2016b) approached continuous robotic control problems with success by applying Maximum Entropy IRL algorithms which are very closely related to GAIL (Finn et al., 2016a) and have similar drawbacks. ",
+        "bbox": [
+            174,
+            700,
+            825,
+            770
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning from Demonstrations: Hester et al. (2018) developed deep Q-Learning from demonstration (DQfD), in which expert trajectories are added to experience replay and jointly used to train agents along with their own experiences. This was later extended by Vecerik et al. (2017) and Pohlen et al. (2018) to better handle sparse-reward problems in control and Atari games respectively. Despite their efficiency, this class of methods still requires access to rewards in order to learn. ",
+        "bbox": [
+            174,
+            776,
+            825,
+            847
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "GAIL: Following the success of Generative Adversarial Networks (Goodfellow et al., 2014) in image generation, GAIL (Ho & Ermon, 2016) applies adversarial learning to the problem of imitation. Although many variants are introduced in the literature (Li et al., 2017; Fu et al., 2018; Merel et al., 2017; Zhu et al., 2018; Baram et al., 2017), making GAIL work for high-dimensional input spaces, particularly raw pixels, remains a challenge. ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A few papers (Peng et al., 2018; Reed et al., 2018; Blondé & Kalousis, 2018) seek to address the problem of overfitting the discriminator. Peng et al. (2018) introduces the Variational Bottleneck to regularize the discriminator. Reed et al. (2018) proposes to not train the vision module of the discriminator (e.g. use the vision module of the critic network instead) and only train a tiny network on top of the vision module to discriminate. Blondé & Kalousis (2018) follow a similar approach. Unstructured regularization, however, cannot stop the discriminator from fitting to features that are systematically different between the agents’ and the demonstration’s behavior, like those illustrated in Figure 2. TRAIL, on the other hand, is much less prone to overfitting to these features. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Stadie et al. (2017) extend GAIL to the setting of third person imitation, in which the demonstrator and agent observations come from different views. To prevent the discriminator from discriminating based on viewpoint domain, they use gradient flipping from an auxiliary classifier to learn domain-invariant features. Our approach is not to learn domain-invariant features, but instead learn domain-agnostic discriminators that only focus on behavior. ",
+        "bbox": [
+            174,
+            222,
+            825,
+            291
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Several recent works have focused on improving the sample efficiency of GAIL (Blondé & Kalousis, 2018; Sasaki et al., 2018). Common to these approaches and to this work, is the use of off-policy actor critic agents and experience replay, to improve the utilization of available experience. ",
+        "bbox": [
+            174,
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+            340
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 REINFORCEMENT LEARNING AND ADVERSARIAL IMITATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            369,
+            709,
+            386
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following the notation of Sutton & Barto (2018), a Markov Decision Process (MDP) is a tuple $( \\mathcal { S } , \\mathcal { A } , R , \\mathbf { \\bar { P } } , \\gamma )$ with states $s$ , actions $\\mathcal { A }$ , reward function $R ( s , a )$ , transition distribution $P ( s ^ { \\prime } | s , a )$ , and discount $\\gamma$ . An agent in state $s \\in S$ takes action $a \\in A$ according to its policy $\\pi$ and moves to state $s ^ { \\prime } \\in \\varDelta$ according to the transition distribution. The goal of RL algorithms is to find a policy that maximizes the expected sum of discounted rewards, represented by the action value function $\\begin{array} { r } { Q ^ { \\pi } ( s , a ) = \\mathbb { E } ^ { \\pi } [ \\sum _ { t = 0 } ^ { \\infty } \\overset { \\cdot } { \\gamma } ^ { t } R ( s _ { t } , a _ { t } ) ] } \\end{array}$ , where $\\mathbb { E } ^ { \\pi }$ is an expectation over trajectories starting from $s _ { 0 } = s$ and taking action $a _ { 0 } = a$ and thereafter running the policy $\\pi$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "To apply RL, it is essential that we have access to the reward function which is often hard to design and evaluate (Singh et al., 2019). In addition, sparse rewards can cause exploration difficulties that pose great challenges to RL algorithms. We therefore look to imitation learning and particularly GAIL to derive a reward function from expert demonstrations. In GAIL, a reward function is learned by training a discriminator network $D ( s , a )$ to distinguish between agent and expert state-action pairs. The GAIL objective is thus formulated as follows: ",
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+        "img_path": "images/108a841f22d52fb14679a7fca453346693df0d23802f2235bfac6b83e1870dfa.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pi } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { ( s , a ) \\sim \\pi _ { E } } [ \\log D ( s , a ) ] + \\mathbb { E } _ { ( s , a ) \\sim \\pi } [ \\log ( 1 - D ( s , a ) ) ] - \\lambda _ { H } H ( \\pi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\pi$ is the agent policy, $\\pi _ { E }$ the expert policy, and $H ( \\pi )$ an (optional) entropy regularizer. The reward function is defined simply: $R ( s , a ) = - \\log ( 1 - D ( s , a ) )$ . ",
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+        "type": "text",
+        "text": "GAIL is theoretically appealing and practically simple. The discriminator, however, can focus on any features to discriminate, whether these features are task-relevant or not. In the next subsection we describe a way to constrain the discriminator network in order to prevent it from using task-irrelevant details to distinguish agent and expert data. ",
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+    {
+        "type": "text",
+        "text": "4 TASK-RELEVANT ADVERSARIAL IMITATION LEARNING (TRAIL) ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We want the discriminator to focus on task-relevant features. Our proposed solution, TRAIL, prevents the discriminator from being able to distinguish expert and agent behaviour based on selected aspects of the data. For instance, the discriminator should distinguish agent and expert frames only when meaningful behavior is present in those frames. In the absence of behavior useful to solve the task, e.g. in initial frames prior to the execution of the behavior, the discriminator should be agnostic. ",
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+        "type": "text",
+        "text": "To this end, we propose a constraint for the discriminator, such that its accuracy must not be greater than chance on a particular set of observations $\\mathcal { L }$ that we will call the invariant set. $\\mathcal { T }$ will include only observations that can be distinguished as agent or expert in task-irrelevant ways. Precisely, we formulate TRAIL in terms of the following constrained optimization problem: ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/677a3c547cb7740be31ef7ad9de87cb3fd4d4b0088c73a1397ca4bd2fd8ee8c4.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\displaystyle \\mathop { \\operatorname* { m a x } } _ { \\psi } } & { \\mathbb { E } _ { s \\sim \\pi _ { E } } \\left[ \\log D _ { \\psi } ( s ) \\right] + \\mathbb { E } _ { s \\sim \\pi _ { \\theta } } \\left[ \\log ( 1 - D _ { \\psi } ( s ) ) \\right] } \\\\ { \\displaystyle s . t . } & { \\frac { 1 } { 2 } \\mathbb { E } _ { s \\sim \\pi _ { E } } \\left[ { \\mathbf 1 } _ { { \\mathbf D } _ { \\psi } ( { \\mathbf s } ) \\geq \\frac { 1 } { 2 } } \\mid s \\in { \\mathbb Z } \\right] + \\frac { 1 } { 2 } \\mathbb { E } _ { s \\sim \\pi _ { \\theta } } \\left[ { \\mathbf 1 } _ { { \\mathbf D } _ { \\psi } ( { \\mathbf s } ) < \\frac { 1 } { 2 } } \\mid s \\in { \\mathbb Z } \\right] \\le \\frac { 1 } { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To apply the above constraint in practice, we can optimize the reverse of the objective function on $\\mathcal { T }$ That is, given a batch of $N$ examples $s _ { e } \\sim \\pi _ { E } , s _ { \\theta } \\sim \\pi _ { \\theta }$ from the expert and agent, and $\\hat { s } _ { e } \\sim \\pi _ { E }$ and $\\hat { s } _ { \\theta } \\sim \\pi _ { \\theta }$ both in the set $\\mathcal { T }$ , we maximize the following augmented objective function: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4d4c40cd8a255e2543089e2861ddce6e8fe5af859ebd3aa65fb0b438c9523f44.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\psi } \\big ( s _ { e } , s _ { \\theta } , \\hat { s } _ { e } , \\hat { s } _ { \\theta } \\big ) = \\sum _ { i = 1 } ^ { N } \\log D _ { \\psi } \\left( s _ { e } ^ { ( i ) } \\right) + \\log \\left( 1 - D _ { \\psi } \\left( s _ { \\theta } ^ { ( i ) } \\right) \\right) } \\\\ & { \\qquad - \\lambda \\left[ \\sum _ { i = 1 } ^ { N } \\log D _ { \\psi } \\left( \\hat { s } _ { e } ^ { ( i ) } \\right) + \\log \\left( 1 - D _ { \\psi } \\left( \\hat { s } _ { \\theta } ^ { ( i ) } \\right) \\right) \\right] \\mathbf { 1 } _ { a c c u r a c y ( \\hat { s } _ { e } , \\hat { s } _ { \\theta } ) \\geq \\frac { 1 } { 2 } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $a c c u r a c y ( \\cdot , \\cdot )$ is defined as the average of discriminator accuracies: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d4fae644884feb2b3543a804ce727543c012afad257cd167eb5487d306ecc872.jpg",
+        "text": "$$\na c c u r a c y ( \\hat { s } _ { e } , \\hat { s } _ { \\theta } ) = \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ { N } \\left[ \\mathbf { 1 } _ { \\mathrm { D } _ { \\psi } \\left( \\hat { \\mathbf { s } } _ { e } ^ { ( \\mathrm { i } ) } \\right) \\geq \\frac { 1 } { 2 } } + \\mathbf { 1 } _ { \\mathrm { D } _ { \\psi } \\left( \\hat { \\mathbf { s } } _ { \\theta } ^ { ( \\mathrm { i } ) } \\right) < \\frac { 1 } { 2 } } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The scalar $\\lambda \\geq 0$ is a tunable hyperparameter. ",
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+    {
+        "type": "text",
+        "text": "We used single frames as states but sequences can also be used, and we do not require expert actions. Not requiring actions enables learning in very off-policy settings, where the action dimensions and distributions of the demonstrator (another robot or human) are different from those of the agent. ",
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+    {
+        "type": "text",
+        "text": "4.1 THE SELECTION OF THE INVARIANT SET ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The selection of the invariant set $\\mathcal { T }$ is a design choice. In general, we could always contrive nonstationary and adversarial ways of making this choice difficult. However, we argue that in many situations of great interest, including our robotic manipulation setup, it is easy to propose effective and very general invariant sets. ",
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+    {
+        "type": "text",
+        "text": "A straightforward way to collect robot data is to execute a random policy. We can then use the resulting random episodes, for both expert and agent, to construct the invariant set $\\mathcal { T }$ . Another way to construct $\\mathcal { T }$ is to use early frames from both expert and agent episodes. Since in early frames little or no task behavior is apparent, this strategy turns out to be effective and no extra data has to be collected. This strategy also improves robustness with respect to variation in the initial conditions of the task; see for example block insertion in Figure 1(d). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Importantly, if the set $\\mathcal { T }$ captures some forms of irrelevance but not all forms, it will nonetheless always help in improving performance. In this regard, TRAIL will dominate its GAIL predecessor whenever the designer has some prior on what aspects of the data might be task irrelevant. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We focus on solving robot manipulation tasks. The environment implements two work-spaces: one with a Kinova Jaco arm (Jaco), and the other with a Sawyer arm (Sawyer). See supplementary material A.1 for a detailed description. Environment rewards, which are not used by GAIL-based methods, are sparse and equal to $+ 1$ for each step when a given task is solved and 0 otherwise. The maximum reward for the episode is 200, since this is the length of a single evaluation episode. ",
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+    {
+        "type": "text",
+        "text": "Our agent is based on the off-policy D4PG algorithm (Barth-Maron et al., 2018) because of its stability and data-efficiency (see supplementary material A.6). Following Vecerik et al. (2017), we add expert demonstrations into the agents’ experience replay, and refer to the resulting RL algorithm as D4PG from Demonstrations (D4PGfD). For each task we collect 100 human demonstrations. ",
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+    {
+        "type": "text",
+        "text": "Data augmentation Traditional data augmentation has proved beneficial in imitation learning (Berseth & Pal, 2019). Surprisingly, to the best of our knowledge, this has not been explicitly studied in prior publications on GAIL. However, we find that data augmentation is a generally useful component to prevent discriminator overfitting. It drastically improves the baseline GAIL agent, and is necessary to solve any of the harder manipulation tasks. We distort images by randomly changing brightness, contrast and saturation; random cropping and rotation; adding Gaussian noise. When multiple sensor inputs are available (e.g. multiple cameras), we also randomly drop out these inputs, but leaving at least one active. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "All discriminator-based methods in this work (including all baselines) use data augmentation except as indicated in the relevant ablations (see Section 5.2). We also considered regularizing the GAIL discriminator with spectral normalization (Miyato et al., 2018). It performed slightly better than GAIL, but still failed in the presence of distractor objects, and we thus omit spectral normalization in the main experiments for simplicity. ",
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+    {
+        "type": "text",
+        "text": "Actor early stopping When the agent has learned the desired behavior, and the resulting data is used for training, the discriminator will become unable to distinguish expert and agent observations based only on behavior. This forces the discriminator to rely on task-irrelevant information. ",
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+    {
+        "type": "text",
+        "text": "To avoid this scenario, we propose to restart each actor episode after a certain number of steps such that successful behavior is rarely represented in agent data. This enables the discriminator to recognize the goal condition, which appears frequently at the end of demonstration episodes, as representative of expert behavior. To avoid hand-tuning the stopping step number, we found that the discriminator score can be used to derive an adaptive stopping criterion. Concretely, we restart an episode if the discriminator score at the current step exceeds the median score of the episode so far for $T _ { p a t i e n c e }$ consecutive steps (in practice we set $T _ { p a t i e n c e } = 1 0$ ). ",
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+    {
+        "type": "text",
+        "text": "We set invariant set hyperparameter $\\lambda = 1$ and use adaptive early stopping for TRAIL. The ablation with $\\lambda = 0$ , i.e. adaptive early stopping only, is referred to as TRAIL-0. Hence, the only difference between TRAIL-0 and TRAIL is that the later uses invariant set constraints. ",
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+    {
+        "type": "text",
+        "text": "5.1 BLOCK LIFTING WITH DISTRACTORS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we consider two variants of the lift task in the Sawyer work space: a) lift alone, where only one red cube is present, b) lift distracted, with two extra blocks (blue and green, see Figure 9). We show how adding these additional distractors affects the training procedure. ",
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+        "type": "text",
+        "text": "We first compare our method to baselines. In doing so, the invariant set is constructed using the first 10 frames from every episode, as in the rest of the work. This choice does not require us to collect any extra data, and hence the comparison with baselines is fair. In the next section, we elaborate on the choice of the invariant set and provide additional experimental results. ",
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+        "type": "text",
+        "text": "As baselines, we run BC and GAIL (with data augmentation). Baseline GAIL is the strongest we were able to implement. The only difference between this baseline and TRAIL is the use of invariant set constrains and actor early stopping (code, agent configurations like number of actors, and network architecture are the same). We additionally consider the approaches proposed by Reed et al. (2018) as GAIL-based baselines; using either a randomly initialized convolutional network, or a convolutional critic network, to provide fixed vision features on top of which a tiny discriminator network is trained. We call these two baselines random and critic respectively. Finally, to show the importance of actor early stopping, we run TRAIL-0 (Fig. 3). ",
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+    {
+        "type": "image",
+        "img_path": "images/59bb3bff9b0dfb40d7b0e4e3cc9fbc548f0f04232c09b94486607c7ff7cfd013.jpg",
+        "image_caption": [
+            "Figure 3: Results for lift alone, lift distracted, and lift distracted seeded. Only TRAIL excels. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "All methods perform satisfactorily on lift alone, but the proposed methods TRAIL-0 and TRAIL do best. As expected, the performance of BC on lift distracted is similar to its performance on lift alone, despite the two additional blocks. The two additional blocks in lift distracted affect the GAIL-based baselines, despite being irrelevant to the task. ",
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+        "type": "text",
+        "text": "To understand this effect, we conducted an additional experiment (lift distracted seeded). Here the initial block positions are randomly drawn from the expert demonstrations. Therefore, it is impossible to discriminate between expert and actor episodes using the first few frames of an episode. Note this initialization procedure is not applied to the evaluation actor, keeping the evaluation scores comparable between lift distracted and lift distracted seeded. ",
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+    {
+        "type": "text",
+        "text": "This experiment exposes one major culprit behind the performance degradation of GAIL: memorization. The discriminator can achieve perfect accuracy by memorizing all 100 initial positions from the demonstration set, making the reward function uninformative. By constraining the discriminator, TRAIL squeezes out this irrelevant information and succeeds in solving the task in the presence of distractions. TRAIL is the only method that is able to handle the variety of initial cube positions during training, achieving better than expert performance on lift distracted. ",
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+    {
+        "type": "text",
+        "text": "Interestingly, random performs reasonably on lift distracted. Given random’s strong performance, we conducted additional experiments to evaluate its effectiveness when trained with adaptive early stopping and present the results in Figure 12 of the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "CONSTRUCTING THE INVARIANT SET $\\mathcal { T }$ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In the previous subsection, early frames were used to construct the invariant set (TRAIL-early). Here, we evaluate another previously mentioned approach for constructing $\\mathcal { T }$ ; random policy (TRAILrandom). The lift distracted task caused all baselines to fail, but was solved by TRAIL. We introduce a harder version of the task, where the expert appearance is different, to tease out the differences between TRAIL-early and TRAIL-random. The difference in appearance between the expert and imitator allows the GAIL discriminator to trivially distinguish them. The results and the differences in the expert appearance are presented in Figure 4. ",
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+    {
+        "type": "image",
+        "img_path": "images/6b8cf409f166f495f4aaf1c532472a37e5e94c105788ca3abc316459da5a2a6d.jpg",
+        "image_caption": [
+            "Figure 4: Lift red block, where expert has a different body appearance, and with distractor blocks. TRAIL-random outperforms GAIL, and performs on par with TRAIL-early. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The new task is indeed harder and it takes longer for TRAIL methods to achieve performance better than BC baseline, which is not affected by the different body appearance. GAIL is clearly outperformed and does not take off. ",
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+    {
+        "type": "text",
+        "text": "The difference between TRAIL methods is negligible. We also tried to mix them but the differences remain imperceptible. Hence, in the following experiments we simply use early frames. This choice is pragmatic as it does not require that we collect any extra data, and hence the comparison with GAIL and other baselines is fair. It is also very simple to apply in practice, even if one does not have access to the expert setup anymore. Finally, it is general and powerful enough to be successfully used across all robotic manipulation tasks considered in this work. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "To decide how many initial frames should be used to construct $\\mathcal { T }$ , we conducted an ablation study and found out that the method is not very sensitive to this choice (see supplementary material A.2). Hence, we chose 10 initial frames, and intentionally used the same number for all tasks to further emphasize generality of this choice. ",
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+    {
+        "type": "text",
+        "text": "5.2 ABLATION STUDIES ",
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+    {
+        "type": "text",
+        "text": "MEASURING DISCRIMINATOR MEMORIZATION OF TASK-IRRELEVANT FEATURES ",
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+        "type": "text",
+        "text": "In this section, we experimentally confirm that memorization is a limiting factor of GAIL methods. We equipped the discriminator with two extra heads whose inputs are the final spatial layer of the ",
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+        "type": "text",
+        "text": "ResNet for the lift distracted task. The first head is trained on the first frames only, and has the same target as the main head (i.e. discriminating between agent and expert). To train the second head, we randomly divide the expert demonstrations into two equi-numerous subsets, and the task is to predict to which of these randomly chosen sets the demonstration was assigned to. Both heads are trained via backpropagation but their gradient is not propagated to the ResNet so they do not influence the training procedure directly. ",
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+    {
+        "type": "text",
+        "text": "If our claim is correct, we expect the extra heads to have higher accuracy for TRAIL-0 as compared to TRAIL, since TRAIL representations are penalized for having features triggering memorization, i.e. the features should not aid discriminating based on the first frames or predicting a random label for each expert demonstration. No reasonably performing method is able to force $50 \\%$ accuracy for extra heads since some features are important to solve the task (e.g. the position of the red cube). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also collected 25 extra holdout demonstrations and visualize the average discriminator prediction on them, and compare with predictions on the training demonstrations. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/f3d0ef0720ecd274b57537e078d92668d28a832dccb1c8bad24328a24d337bc8.jpg",
+        "image_caption": [
+            "Figure 5: Demonstrating the memorization problem on the lift distracted task (here higher accuracy is worse). Accuracy of different discriminator heads is presented (A-D). In A, the overall accuracy for all timesteps. Then main $\\mathrm { ( m ) }$ and extra (e) heads accuracy for the first steps are presented in B and C, respectively. Accuracy of the head predicting randomly assigned demonstration class is shown in D. Average discriminator predictions for training and holdout demonstration are shown in E and F. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            316,
+            797,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 5, we see the overall accuracy of the main head for TRAIL-0 is significantly higher, and the difference is larger for early steps (when TRAIL achieves only $50 \\%$ , as expected). TRAIL representations are less helpful for extra heads (see Figure 5 C-D). Finally, TRAIL-0 clearly overfits on training demonstrations, predicting almost the maximum score, while only $\\sim 0 . 2 5$ is predicted for holdout demonstrations. The TRAIL average predictions for both datasets are almost identical. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ACTOR EARLY STOPPING (TRAIL-0) ",
+        "bbox": [
+            176,
+            708,
+            431,
+            722
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we analyze the importance of adaptive early stopping on 3 tasks in the Jaco workspace: lift red cube $( l i f t )$ , put red cube in box $( b o x )$ , and stack red cube on blue cube (stack). We consider D4PGfD and three GAIL-based models with varying termination policies: a) fixed step (50), b) based on ground truth task rewards, and c) based on adaptive early stopping (TRAIL-0). Using ground truth task rewards, an episode is terminated if the reward at the current step exceeds the median reward of the episode so far for 10 consecutive steps. Results are presented in Figure 6. ",
+        "bbox": [
+            174,
+            734,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Termination based on task reward is clearly superior; although unrealistic in practice, it defines the performance upper-bound and clearly shows that early stopping is beneficial. TRAIL-0 is robust and reaches human performance on all tasks. A fixed termination policy, when tuned, can be very effective. The same fixed termination step, however, does not work for all tasks. See Figure 13 in supplementary material for the effects of varying termination steps. Finally, as can be inferred from stack results, the dense rewards provided by TRAIL-0 are helpful in solving this challenging problem which is unsolved with D4PGfD even though D4PGfD uses ground truth task rewards. ",
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/236f966ae10dec709f176a7097a72d5e3722052c12c4e9e616a6e0ea34d38c6a.jpg",
+        "image_caption": [
+            "Figure 6: Results for lift, box, and stack on Jaco environment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            94,
+            805,
+            212
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "DATA AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            266,
+            330,
+            280
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Table 5.2, we report the best reward obtained in the first 12 hours of training (averaged for all seeds;   \nsee Figure 14 for full curves). The results show that data augmentation is needed in lift distracted.   \nFor the lift alone task, TRAIL-0 with data augmentation performs on par with TRAIL. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/44e435f0cddc322fb45ee6a3331dd739fc71ea12dd406b3c24ab878b32347dd0.jpg",
+        "table_caption": [
+            "Table 1: Influence of data augmentation (evaluated on rewards) for lift alone and lift distracted. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>Regularization</td><td>Data augmentation</td><td>No data augmentation</td></tr><tr><td>lift alone</td><td>TRAIL-0 TRAIL</td><td>~165</td><td>~115</td></tr><tr><td rowspan=\"2\">lift distracted</td><td>TRAIL-0</td><td>~155</td><td>~165</td></tr><tr><td>TRAIL</td><td>~30 ~180</td><td>~5 ~10</td></tr></table>",
+        "bbox": [
+            253,
+            354,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The performance of TRAIL is not affected by the lack of data augmentation on the lift alone task, whereas the performance of TRAIL-0 is. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "LEARNING WITH A FIXED, PERFECT DISCRIMINATOR ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            532,
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+            546
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To assess whether learned discriminators are necessary, we compare TRAIL against agents using a fixed reward function corresponding to $R _ { e x p e r t } = 1$ and $R _ { a g e n t } = 0$ for the lift alone and lift distracted tasks. This baseline simulates an oracle discriminator with perfect generalization, but which is agnostic to behavior. On the lift alone task, agents using this fixed reward achieve roughly half the reward of TRAIL asymptotically, and on lift distracted they do not solve the task (average rewards are less than 5). See supplementary Figure 16 for learning curves. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 LEARNING FROM OTHER EMBODIMENTS AND PROPS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            667,
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+            680
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Since the TRAIL discriminator is trained to ignore task-irrelevant features, it can learn from demonstrations with different embodiments and props. Figure 7 shows that GAIL even with augmentation fails to learn block lifting from a different embodiment, and performs worse when the expert uses a different prop color. TRAIL solves the task and achieves better performance in both cases. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c312fba1da06c95e98c0f3e62ac762e79d2ef8e05c275905eb6accf57d14aeb2.jpg",
+        "image_caption": [
+            "Figure 7: When the expert differs in body or prop appearance, TRAIL outperforms GAIL. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            771,
+            792,
+            887
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 EVALUATION ON DIVERSE MANIPULATION TASKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
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+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To further demonstrate benefits of using our proposed method, we present results for TRAIL, TRAIL0, the baseline GAIL, and BC on more challenging tasks. Specifically, we consider stack with the Sawyer robot; and insertion and stack banana in the Jaco work-space. The results are shown in Figure 8. The tasks we consider here are much harder as evidenced by the performance of BC agents. These experiments suggest that TRAIL is generally useful as an improvement over GAIL, even when the tasks are not designed to include task-irrelevant information. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/b34a66b68ab4b752173566a72eef0ab16bcf79a532090242dd93629c664709c9.jpg",
+        "image_caption": [
+            "Figure 8: Results comparing TRAIL, TRAIL-0 and GAIL for diverse manipulation tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To make adversarial imitation work on nontrivial tasks from pixels, it is crucial to prevent the discriminator from exploiting task-irrelevant information. Data augmentation is a necessary component that prevents the discriminator from overfitting, but alone is not sufficient to solve more challenging problems. Our proposed method TRAIL proved to be a remedy which effectively focuses the discriminator on the task even when task-irrelevant features are present. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "We showed through ablations that TRAIL benefits from data augmentation, actor early stopping, and the use of invariant set constraints. The last enables our agent to solve the full task suite in our experiments showing its decisive importance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
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+        "text": "Avi Singh, Larry Yang, Kristian Hartikainen, Chelsea Finn, and Sergey Levine. End-to-end robotic reinforcement learning without reward engineering. arXiv preprint arXiv:1904.07854, 2019. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Bradly C Stadie, Pieter Abbeel, and Ilya Sutskever. Third-person imitation learning. arXiv preprint arXiv:1703.01703, 2017. ",
+        "bbox": [
+            173,
+            315,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5026–5033. IEEE, 2012. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Instance normalization: The missing ingredient for fast stylization. arXiv preprint arXiv:1607.08022, 2016. ",
+        "bbox": [
+            173,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Matej Vecerik, Todd Hester, Jonathan Scholz, Fumin Wang, Olivier Pietquin, Bilal Piot, Nicolas Heess, Thomas Rothörl, Thomas Lampe, and Martin A. Riedmiller. Leveraging demonstrations for deep reinforcement learning on robotics problems with sparse rewards. CoRR, 2017. ",
+        "bbox": [
+            173,
+            443,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yuke Zhu, Ziyu Wang, Josh Merel, Andrei Rusu, Tom Erez, Serkan Cabi, Saran Tunyasuvunakool, János Kramár, Raia Hadsell, Nando de Freitas, et al. Reinforcement and imitation learning for diverse visuomotor skills. In RSS, 2018. ",
+        "bbox": [
+            174,
+            491,
+            823,
+            530
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, and Anind K Dey. Maximum entropy inverse reinforcement learning. In AAAI, volume 8, pp. 1433–1438. Chicago, IL, USA, 2008. ",
+        "bbox": [
+            169,
+            540,
+            823,
+            566
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            447,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 DETAILED DESCRIPTION OF ENVIRONMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            132,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "All our simulations are conducted using $\\mathbf { M u J o C o ^ { 1 } }$ (Todorov et al., 2012). We test our proposed algorithms in a variety of different envrionments using simulated Kinova Jaco2, and Sawyer robot arms3; see Figure 9. We use the Robotiq 2F85 gripper4 in conjuction with the Sawyer arm. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/6bccf9982d118787a6c2979868292057dd3b8d908253da25a031063cb76c20dc.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            212,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 9: Two work spaces, Jaco (left) which uses the Jaco arm and is $2 0 \\times 2 0 ~ \\mathrm { c m }$ , and Sawyer (right) which uses the Sawyer arm and more closely resembles a real robot cage and is $3 5 \\times 3 5 \\mathrm { c m }$ . ",
+        "bbox": [
+            176,
+            338,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To provide demonstrations, we use the SpaceNavigator 3D motion controller5 to set Cartesion velocities of the robot arm. The gripper actions are implemented via the buttons on the controller. All demonstrations in our experiments are provided via human teleoperation and we collected 100 demonstrations for each experiment. ",
+        "bbox": [
+            174,
+            380,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jaco: When using the Jaco arm, we use joint velocity control (9DOF) where we control all 6 joints of arm and all 3 joints of the hand. The simulation is run with a numerical time step of 10 milliseconds, integrating 5 steps, to get a control frequency of 20HZ. The agent uses a frontal camera of size $6 4 \\times 6 4$ (see Figure 10(a)). For a full list of observations the agent sees, please refer to Table 2(a). ",
+        "bbox": [
+            174,
+            444,
+            826,
+            500
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/b6bc13e4b98119453319a74daf7dde365754b163a2a6e1e98dd817c38debb41d.jpg",
+        "image_caption": [
+            "Figure 10: Illustration of the pixels inputs to the agent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            511,
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+            664
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sawyer: When using the Sawyer arm, we use Cartesian velocity control (6DOF) for the robot arm and add one additional action for the gripper resulting in 7 degrees of freedom. The simulation is run with a numerical time step of 10 milliseconds, integrating 10 steps, to get a control frequency of 10HZ. The agent uses two frontal cameras of size $6 4 \\times 6 4$ situated on the left and right side of the robot cage respectively (see Figure $1 0 ( \\mathrm { b } , \\mathrm { c } ) ,$ ). For a full list of observations the agent sees, please refer to Table 2(b). ",
+        "bbox": [
+            173,
+            705,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For all environments considered in this paper, we provide sparse rewards (i.e. if task is accomplished, the reward is 1 and 0 otherwise). In experiments regards our proposed methods, rewards are only used for evaluation purposes and not for training the agent. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/07b4e70c00c60a1551189151f9ee412f3336bc9f65211675aee4c92bca3c9287.jpg",
+        "table_caption": [
+            "a) Jaco "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Feature Name</td><td>Dimensions</td></tr><tr><td>frontal camera base force and torque sensors</td><td>64×64×3 6</td></tr><tr><td>arm joints position</td><td>6</td></tr><tr><td>arm joints velocity</td><td>6</td></tr><tr><td>wrist force and torque sensors</td><td>6</td></tr><tr><td>hand finger joints position</td><td>3</td></tr><tr><td>hand finger joints velocity</td><td></td></tr><tr><td>hand fingertip sensors</td><td>3</td></tr><tr><td></td><td>3</td></tr><tr><td>grip site position</td><td>3</td></tr><tr><td>pinch site position</td><td>3</td></tr></table>",
+        "bbox": [
+            169,
+            133,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/35828ddf5a5369eb1bb42ba00067e3cdf43e97fe28c656aea544c66d4b4046ca.jpg",
+        "table_caption": [
+            "Table 2: Observation and dimensions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">b) Sawyer</td></tr><tr><td>Feature Name frontleft camera</td><td>Dimensions 64×64×3</td></tr><tr><td>front right camera</td><td>64×64×3</td></tr><tr><td>arm joint position</td><td>7</td></tr><tr><td>arm joint velocity</td><td>7</td></tr><tr><td>wrist force sensor</td><td>3</td></tr><tr><td>wrist torque sensor</td><td>3</td></tr><tr><td>hand grasp sensor</td><td></td></tr><tr><td></td><td>1</td></tr><tr><td>hand joint position</td><td>1</td></tr><tr><td>tool center point cartesian orientation</td><td>9</td></tr><tr><td>tool center point cartesian position</td><td></td></tr><tr><td>hand joint velocity</td><td>3 1</td></tr></table>",
+        "bbox": [
+            480,
+            126,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 EARLY FRAMES ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            321,
+            452,
+            335
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here, we vary the number of early frames of each episode used to form $\\mathcal { T }$ . We report that a range of values from 1 up to 20 works well across both tasks (Put in box and Stack banana), with performance gradually degrading as the value increased beyond 20 (Fig. 11). ",
+        "bbox": [
+            174,
+            348,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/62c162c76431293f01240e50adc3fc2894bfed770d528775da6f48cfa37dd471.jpg",
+        "image_caption": [
+            "Figure 11: TRAIL performance, varying the number of first frames in each episode used to form $\\mathcal { T }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            406,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 TRAIL-0 WITH random ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            582,
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+            597
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We found out and mentioned in the subsection 5.1 that random benefits from TRAIL-0. Unfortunately, it is still prone to overfitting and hence, worse than our full method – TRAIL. We present random $^ +$ TRAIL-0 accompanied with our methods in Figure 12. Our TRAIL and random $^ +$ TRAIL-0 are the only methods exceeding BC performance on lift distracted. However, TRAIL performance is clearly better (obtains higher rewards and never gets overfitted). ",
+        "bbox": [
+            173,
+            608,
+            825,
+            679
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed344e1142f29ef99b9a64add4b50040c0bf4ab9992f0071804687203c06a46a.jpg",
+        "image_caption": [
+            "Figure 12: Results for lift alone, lift distracted, and lift distracted seeded. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            693,
+            803,
+            801
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 FIXED TERMINATION POLICY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
+            419,
+            856
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As mentioned in the subsection 5.2, the most basic early termination policy – fixed step termination – may be very effective if tuned. Since the tuning may be expensive in practice, we recommend using adaptive early stopping (TRAIL-0). However, for the sake of completeness we provide results for fixed step termination policy depending on the hyperparameter tuned. The results for stack task are presented in Figure 13. The Jaco work space is considered here because Sawyer requires TRAIL to obtain high rewards. As can be inferred from the figure, the performance is very sensitive to the fixed step hyperparameter. We refer to subsection 5.2 for more comments on all methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/6ce8e8947b35ee83aff9d74c4c2c443c36dcd3ab4361b7b7f06e4556241f9585.jpg",
+        "image_caption": [
+            "Figure 13: Results for stack in Jaco work space. Fixed step termination policy can be very effective but the final performance is very sensitive to the hyperparameter. TRAIL-0 does not need tuning nor access to the environment reward. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            310,
+            160,
+            676,
+            309
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 DATA AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            388,
+            374,
+            402
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "An extra set of experiments on lift alone and lift distracted tasks (described in subsection 5.1) has been performed to show importance of data augmentation. ",
+        "bbox": [
+            173,
+            414,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Because in the subsection 5.2 only the peak performance is presented (Table 5.2), we present here the full curves in Figure 14. The results shows that data augmentation is necessary to obtain high rewards in lift distracted. For easier lift alone, TRAIL-0 with data augmentation perform at par with TRAIL. ",
+        "bbox": [
+            174,
+            450,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb24dedec05c204244af33e7fd3602ca1d47b8b427d2224cef1cd3839e7e72d0.jpg",
+        "image_caption": [
+            "Figure 14: Results for lift alone and lift distracted in Sawyer work space. TRAIL and TRAIL-0 are by default with data augmentation. Additional results for these methods without data augmentation are presented to show its importance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            506,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6 D4PG ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            737,
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+            751
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We use D4PG (Barth-Maron et al., 2018) as our main training algorithm. Briefly, D4PG is a distributed off-policy reinforcement learning algorithm for continuous control problems. In a nutshell, D4PG uses Q-learning for policy evaluation and Deterministic Policy Gradients (DPG) (Silver et al., 2014) for policy optimization. An important characteristic of D4PG is that it maintains a replay memory $\\mathcal { M }$ (possibility prioritized (Horgan et al., 2018)) that stores SARS tuples which allows for off-policy learning. D4PG also adopts target networks for increased training stability. In addition to these principles, D4PG utilized distributed training, distributional value functions, and multi-step returns to further increase efficiency and stability. In this section, we explain the different ingredients of D4PG. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D4PG maintains an online value network $Q ( s , a | \\theta )$ and an online policy network $\\pi ( s | \\phi )$ . The target networks are of the same structures as the value and policy network, but are parameterized by different parameters $\\theta ^ { \\prime }$ and $\\phi ^ { \\prime }$ which are periodically updated to the current parameters of the online networks. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Given the $Q$ function, we can update the policy using DPG: ",
+        "bbox": [
+            173,
+            103,
+            565,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d9a9a466471a108f418f609eafa9bc0cc7366850e41f8714acb68e7b0816f19e.jpg",
+        "text": "$$\n\\mathcal { I } ( \\phi ) = \\mathbb { E } _ { s _ { t } \\sim \\mathcal { M } } \\left[ \\nabla _ { \\phi } Q ( s _ { t } , \\pi ( s _ { t } | \\phi ) | \\theta ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            131,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Instead of using a scalar $Q$ function, D4PG adopts a distributional value function such that $Q ( s _ { t } , a | \\theta ) ~ = ~ \\bar { \\mathbb { E } } \\big [ Z ( s _ { t } , a | \\theta ) \\big ]$ where $Z$ is a random variable such that $Z ~ = ~ z _ { i }$ w.p. $\\qquad p _ { i } \\propto$ $\\exp ( \\omega ( s _ { t } , a | \\theta ) )$ . The $z _ { i }$ ’s take on $V _ { b i n s }$ discrete values that ranges uniformly between $V _ { m i n }$ and $V _ { m a x }$ such that $\\begin{array} { r } { z _ { i } = V _ { m i n } + i \\frac { V _ { m a x } - V _ { m i n } } { V _ { b i n s } } } \\end{array}$ for $i \\in \\{ 0 , \\cdots , V _ { b i n s } - 1 \\}$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To construct a bootstrap target, D4PG uses $\\mathbf { N } .$ -step returns. Given a sampled tuple from the replay memory: $s _ { t } , a _ { t } , \\{ r _ { t } , r _ { t + 1 } , \\cdot \\cdot \\cdot , r _ { t + N - 1 } \\} , s _ { t + N }$ , we construct a new random variable $Z ^ { \\prime }$ such that Z0 = zi + PN−1n=0 w.p. $p _ { i } \\propto \\exp ( \\omega ( s _ { t + N } , \\pi ( s _ { t + N } | \\phi ^ { \\prime } ) | \\theta ^ { \\prime } ) )$ . Notice, $Z ^ { \\prime }$ no longer has the same support. We therefore adopt the same projection $\\Phi$ employed by Bellemare et al. (2017). The training loss for the value function ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4325c14ac5c5caff010358a3b8f4b5186d28590f7cd88b92da06c98b8b3dfb6a.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta ) = \\mathbb { E } _ { s _ { t } , a _ { t } , \\{ r _ { t } , \\cdots , r _ { t + N - 1 } \\} , s _ { t + N } \\sim \\mathcal { M } } \\big [ H \\big ( \\Phi ( Z ^ { \\prime } ) , Z ( s _ { t } , a _ { t } | \\theta ) \\big ) \\big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            324,
+            707,
+            343
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $H$ is the cross entropy. ",
+        "bbox": [
+            174,
+            356,
+            367,
+            371
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D4PG is also distributed following Horgan et al. (2018). Since all learning processes only rely on the replay memory, we can easily decouple the ‘actors’ from the ‘learners’. D4PG therefore uses a large number of independent actor processes which act in the environment and write data to a central replay memory process. The learners could then draw samples from the replay memory for learning. The learner also serves as a parameter server to the actors which periodically update their policy parameters from the learner. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we always have access to expert demonstrations. We, therefore adopt the practice from DQfD and DDPGfD and put the demonstrations into our replay buffers. For more details see Algorithms 1, 2. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Actor ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Given: • an experience replay memory $\\mathcal { M }$   \nfor $n _ { e p i s o d e s }$ do for $t = 1$ to $T$ do Sample action from task policy: $\\boldsymbol { a } _ { t } \\gets \\pi ( \\boldsymbol { s } _ { t } )$ Execute action $a _ { t }$ and observe new state $s _ { t + 1 }$ , and reward $r _ { t }$ Store transition $( s _ { t } , a _ { t } , r _ { t } , s _ { t + 1 } )$ in memory $\\mathcal { M }$ end for   \nend for ",
+        "bbox": [
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+            551,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Learner ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Given: ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• an off-policy RL algorithm A • a replay buffer $\\mathcal { M }$ • a replay buffer of expert demonstrations $\\mathcal { M } _ { e }$ ",
+        "bbox": [
+            232,
+            763,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Initialize A   \nfor $n _ { u p d a t e s }$ do Sample transitions $\\left( { { s _ { t } } , { a _ { t } } , { r _ { t } } , { s _ { t + 1 } } } \\right)$ from $\\mathcal { M }$ to make a minibatch $B$ . Sample transitions $\\left( { { s _ { t } } , { a _ { t } } , { r _ { t } } , { s _ { t + 1 } } } \\right)$ from $\\mathcal { M } _ { e }$ enlarge the minibatch $B$ . Perform a actor update step with Eqn. (5). Perform a critic update step with Eqn. (6). Update the target actor/critic networks every $k$ steps.   \nend for ",
+        "bbox": [
+            189,
+            814,
+            681,
+            917
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.7 NETWORK ARCHITECTURE AND HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
+            581,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Actor and critic share a residual pixel encoder network with eight convolutional layers (3x3 convolutions, three 2-layer blocks with 16, 32, 32 channels), instance normalization (Ulyanov et al., 2016) and exponential linear units (Clevert et al., 2015) between layers. ",
+        "bbox": [
+            174,
+            131,
+            826,
+            172
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The policy is a 3-layer MLP with ReLU activations with hidden layer sizes (300, 200). The critic is a 3-layer MLP with ReLU activations with hidden layer sizes (400, 300). For a illustration of the network. Please see Figure 15. ",
+        "bbox": [
+            173,
+            179,
+            825,
+            223
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/28c5f55c9a59fb03ff40780ca1b034a37b8cd5bc60d7b421bfe54857e6457a48.jpg",
+        "image_caption": [
+            "Figure 15: Network architecture for the policy and critic. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            237,
+            820,
+            364
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The discriminator network uses a pixel encoder of the same architecture as the actor critic, followed by a 3-layer MLP with ReLU activations and hidden layer sizes (32, 32). ",
+        "bbox": [
+            173,
+            420,
+            823,
+            449
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/8d79a56e3510372616f21696af491774a0fb484005300365e419100ca1e2c249.jpg",
+        "table_caption": [
+            "Table 3: Hyper parameters used in robot manipulation experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameters</td><td>Values</td></tr><tr><td>Actor/Critic Input Width</td><td>64</td></tr><tr><td>Actor/Critic Input Height</td><td>64</td></tr><tr><td>D4PG Parameters</td><td></td></tr><tr><td>Vmin</td><td>-50</td></tr><tr><td>Vmax</td><td>150</td></tr><tr><td>Vbins</td><td>21</td></tr><tr><td>N step</td><td>1</td></tr><tr><td>Actor learning rate</td><td>10-4</td></tr><tr><td>Critic learning rate</td><td>10-4</td></tr><tr><td>Optimizer</td><td>Adam (Kingma &amp; Ba (2014))</td></tr><tr><td>Batch size</td><td>256</td></tr><tr><td>Target update period</td><td>100</td></tr><tr><td>Discount factor (y)</td><td>0.99</td></tr><tr><td>Replay capacity</td><td>106</td></tr><tr><td>Number of actors</td><td>32 or 128</td></tr><tr><td>Imitation Parameters</td><td></td></tr><tr><td>Discriminator learning rate</td><td>10-4</td></tr><tr><td>Discriminator Input Width</td><td>48</td></tr><tr><td>Discriminator Input Height</td><td>48</td></tr></table>",
+        "bbox": [
+            279,
+            478,
+            714,
+            786
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.8 COMPARING AGAINST LEARNING WITH FIXED REWARDS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            840,
+            607,
+            854
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To check whether learned discriminators are needed, we compare TRAIL against agents using a fixed reward function corresponding to $R _ { e x p e r t } = 1$ and $R _ { a g e n t } = 0$ for the lift alone and lift distracted tasks. This baseline simulates an oracle discriminator with perfect generalization, but which is agnostic to behavior. On the lift alone task, agents using this fixed reward achieve roughly half the reward of TRAIL asymptotically, and on lift distracted they do not solve the task (average rewards are less than 5). The learning curves are presented in Figure 16. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/73412ac4784215fb9164009bf367fa91028931791c266c531f38108dcd112e50.jpg",
+        "image_caption": [
+            "Figure 16: With fixed rewards, the agent is able to learn lift alone somewhat, but performs worse than TRAIL. When distractor blocks are added, the fixed reward agent fails to learn completely. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            294,
+            146,
+            692,
+            295
+        ],
+        "page_idx": 16
+    }
+]

parse/train/S9ZyhWC17wJ/S9ZyhWC17wJ.md

@@ -0,0 +1,264 @@
+# Learning with Noisy Correspondence for Cross-modal Matching
+
+Guocheng Niu Baidu Inc., China niuguocheng@baidu.com
+
+Zhenyu Huang∗ College of Computer Science Sichuan University, China zyhuang.gm@gmail.com
+
+Xiao Liu TAL Education Group liuxiao15@tal.com
+
+Wenbiao Ding TAL Education Group dingwenbiao@tal.com
+
+Xinyan Xiao Baidu Inc., China xiaoxinyan@baidu.com
+
+Hua Wu Baidu Inc., China wu_hua@baidu.com
+
+Xi Peng† College of Computer Science Sichuan University, China pengx.gm@gmail.com
+
+# Abstract
+
+Cross-modal matching, which aims to establish the correspondence between two different modalities, is fundamental to a variety of tasks such as cross-modal retrieval and vision-and-language understanding. Although a huge number of crossmodal matching methods have been proposed and achieved remarkable progress in recent years, almost all of these methods implicitly assume that the multimodal training data are correctly aligned. In practice, however, such an assumption is extremely expensive even impossible to satisfy. Based on this observation, we reveal and study a latent and challenging direction in cross-modal matching, named noisy correspondence, which could be regarded as a new paradigm of noisy labels. Different from the traditional noisy labels which mainly refer to the errors in category labels, our noisy correspondence refers to the mismatch paired samples. To solve this new problem, we propose a novel method for learning with noisy correspondence, named Noisy Correspondence Rectifier (NCR). In brief, NCR divides the data into clean and noisy partitions based on the memorization effect of neural networks and then rectifies the correspondence via an adaptive prediction model in a co-teaching manner. To verify the effectiveness of our method, we conduct experiments by using the image-text matching as a showcase. Extensive experiments on Flickr30K, MS-COCO, and Conceptual Captions verify the effectiveness of our method. The code could be accessed from www.pengxi. me.
+
+# 1 Introduction
+
+As one of the most fundamental techniques in multimodal learning, cross-modal matching aims to bridge different modalities. In recent years, some cross-modal matching methods [19, 11, 7, 26] have been proposed based on Deep Neural Networks (DNNs), which achieved remarkable progress in a variety of applications, such as clustering [29, 24], image/video captioning [1, 44, 22], cross-modal retrieval [40, 19, 13], and visual question answering [9].
+
+In general, most existing cross-modal matching methods embed different modalities into a common space wherein the similarity of positive cross-modal pairs is maximized and that of the negative ones is minimized. Although these methods have achieved promising results, their success depends on an implicit data assumption, i.e., the training data are correctly aligned across modalities. For example, in the vision-and-language tasks, the text needs to accurately describe the image content, and vice versa. In practice, however, it is extremely expensive and time-consuming to annotate or collect such data pairs. Especially, considering the data collected from the Internet [35, 14], it is inevitable to collect some mismatched pairs which are wrongly treated as the matched ones. To the best of our knowledge, such a special noisy label (correspondence) problem has been ignored so far, which will remarkably degrade the performance of matching methods as shown in our experiments.
+
+![](images/8fe147a50fe0a4e139ee18a498d586b8c26880aa0b567f8e71692812be7e2a62.jpg)  
+Figure 1: Noisy labels vs. Noisy Correspondence. We denote the noisy samples with red lines and clean samples with green lines. The traditional noisy labels mainly refer to the errors in category labels, while the noisy correspondence refers to the alignment errors in paired data. For the noisy correspondence in cross-modal matching, the true positive pair correctly guides the cross-modal matching, while the false positive pair causes incorrect supervision for training.
+
+Based on the above observation, we reveal a new paradigm for the noisy labels, named noisy correspondence. Different from the traditional noisy labels, the noisy correspondence refers to the alignment errors in paired data rather than the errors in category annotations (see Fig. 1). To the best of our knowledge, there is no effort has been devoted to study this new problem and the closest paradigm might be the partially view-aligned problem (PVP) [12, 41]. However, PVP is remarkably different from noisy correspondence, and the latter is more practical than the former. To be specific, PVP focuses on that the cross-modal alignment is totally unavailable, whereas the noisy correspondence focuses on that some correspondences are incorrect. In addition, PVP assumes that some correctly aligned data are available for training, whereas our noisy correspondence assumes that the clean and noisy data are mixed.
+
+To solve the noisy correspondence problem in cross-modal matching, we propose a novel method, named Noisy Correspondence Rectifier (NCR). Our method is based on the memorization effect of DNNs observed in [3, 39], i.e., DNNs tend to learn the simple patterns before fitting noisy samples. Motivated by this empirical observation, NCR divides the data into two relative accurate data partitions, i.e., “noisy” and “clean” subsets, based on their loss difference. After that, NCR employs an adaptive prediction function for label rectifying so that the false positives and the true positives could be identified from the “clean” and the “noisy” subsets, respectively. Furthermore, we propose a novel triplet loss for robust cross-modal matching by recasting the rectified labels as the soft margin.
+
+The main contributions and novelties of this paper could be summarized as below. i) We reveal a new problem in cross-modal analysis, which is also a new paradigm for noisy labels, termed noisy correspondence. Different from the traditional noisy labels, the noisy correspondence refers to the alignment errors in paired data instead of the errors in category annotations. To the best of our knowledge, this work could be the first study on this problem. ii) To solve the noisy correspondence problem, we propose a novel method for learning with noisy correspondence, named Noisy Correspondence Rectifier (NCR). One major novelty of NCR is that the rectified label is elegantly recasted as the soft margin of a triplet loss so that the robust cross-modal matching could be achieved. iii) To verify the effectiveness of our method, we conduct experiments on the image-text matching task. Extensive experiments on three challenging datasets verify the effectiveness of our method in the synthesized and real noises.
+
+# 2 Related works
+
+In this section, we briefly introduce some recent developments in cross-modal matching and learning with noisy labels.
+
+# 2.1 Cross-modal Matching
+
+Most existing cross-modal matching works seek to learn a common space wherein different modalities are comparable. In general, existing works could be roughly divided into two categories: 1) Coarsegrained Matching. It often utilizes multiple neural networks to compute a global feature and each network is used for a specific modality [17, 37, 8]. For example, Kiros et al. [17] use a Convolutional Neural Network (CNN) and a Gated Recurrent Unit (GRU) to obtain the image and text features, while enforcing the similarity of positive pairs larger than that of the negative ones. To further boost the matching performance, ${ \mathrm { V S E } } { + } { + }$ [8] uses some representative negatives to improve the discrimination of the model. 2) Fine-grained Matching. It seeks to measure the fine-grained similarity for cross-modal matching [19, 21, 7]. For example, SCAN [19] proposes learning the latent semantic correspondence between the image regions and words that are extracted by bottom-up attention [1] and GRU, respectively. VSRN [21] adopts a graph convolutional network for semantic reasoning. SGRAF [7] proposes constructing a similarity graph to reason the similarity and adopting an attention filtration technique to eliminate the less-meaningful alignments. Recently Chun et al. [6] introduce a new paradigm for cross-modal matching, i.e. possible many-to-many correspondence that existed in the image and captions. To achieve this, they propose to use probabilistic representations to model the possible one-to-many correspondence.
+
+Although promising results have been achieved in recent years, the existing methods heavily rely on the correctly aligned data. In practice, however, such well-matched data is expensive and timeconsuming to collect. Moreover, some recent works [35, 14] show that a large-scale dataset collected from the wild could remarkably improve the performance of the model. However, such a data will inevitably contain some mismatched pairs. Hence, it is highly expected to develop some methods which are robust against the noisy correspondence, which has not been studied as far as we know. Different from the many-to-many correspondence [6] between image and captions, NCR reveals the noisy correspondence problem which refers to the alignment errors of image-text pairs and proposes to eliminate the negative impact from noisy pairs for downstream tasks.
+
+# 2.2 Learning with Noisy Labels
+
+To handle the possible noisy annotations in the training data, a large number of methods have been proposed and almost all of them focus on the classification task [36, 27]. To reduce the negative impact of the noisy labels, the existing works often resort to robust architecture design, regularization, loss adjustment, or sample selection methods. Here, we mainly introduce the last two approaches which are most related to this work. To be specific, the loss adjustment achieves robustness by adjusting the contribution of clean and noisy samples w.r.t. the loss. For example, Reed et al. [32] proposed a bootstrapping loss based on the model predictions for loss correction. Zhang et al. [45] provided some theoretical explanations for the label correction along with a new label correction algorithm. Different from the loss adjustment methods, sample selection methods aim to select clean samples from a noisy dataset. For example, Arpit et al. [3] showed that DNNs tend to learn simple patterns before fitting noisy samples, namely the memorization effect. Motivated from this, Arazo et al. [2] proposed treating the samples with small loss as the clean samples. To avoid the selection bias of clean samples, Co-teaching methods [10, 43] use the samples with small loss to iteratively train two networks. In recent, DivideMix [20] adopts the MixMatch method [4] for semi-supervised learning with the clean and noisy samples.
+
+Unlike the above noisy label studies, this paper focuses on the noisy correspondence problem which considers mismatched multimodal data pairs instead of incorrectly annotated data points. Besides the difference in the problem, this work is also different from the aforementioned studies in the methodology. To be specific, in cross-modal matching, it is impossible to directly adopt these noise label learning methods to solve the noisy correspondence problem due to the following two reasons. First, most of the noisy label learning methods propose to use the model’s prediction for label rectifying in the scenario of classification, while it is intractable to directly predict the correspondence of given pairs in matching models. Second, even if we can rectify the noisy correspondence somehow, the rectified real-valued labels are incompatible with the existing matching methods since most of them assume the given labels are binary. To address these problems, NCR proposes an adaptive prediction function and a novel triplet loss by recasting the soft labels as soft margins.
+
+![](images/5c322d44fac00fe31e4ff1c8ce4a4a36193b7016a6b16a32f08f9c75ab7a153c.jpg)  
+Figure 2: Overview of the proposed method. (a) Training pipeline of NCR. NCR consists of two individual networks (A, B) which work in the manner of co-teaching. In brief, NCR first warmup the networks (A, B) on the original training data using the loss $L _ { w }$ which is also used for per-sample loss computation. Then, based on the memorization effect of DNclean and noisy subsets at each epoch using either A or B, i.e., $S ^ { A } = ( S _ { c } ^ { A } , S _ { n } ^ { A } )$ the and $S ^ { B } = \mathsf { \bar { ( } } S _ { c } ^ { B } , S _ { n } ^ { B } )$ After that, NCR will co-rectify the correspondence of $\{ { \cal S } ^ { A } , { \cal S } ^ { B } \}$ and obtain $\{ \hat { S ^ { A } } , \hat { S ^ { B } } \}$ using an adaptive prediction function. Finally, $\hat { \cal S } ^ { A }$ and $\hat { S ^ { B } }$ will be used to train the network $B$ and $A$ in a swapping way. (b) Robust image-text matching network. For example, the network A projects the image and text by the modal-specific networks $f$ and $g$ , respectively. Then the similarity $S ( I , T )$ is computed on the extracted features $f ( I )$ and $g ( T )$ . To achieve robust image-text matching, the rectified soft labels are recast as the soft margin of our loss $\scriptstyle L _ { s o f t }$ . As shown, for a given anchor, $\boldsymbol { L _ { s o f t } }$ will enforce the true positive to closer to it than the negative by a large margin $\hat { \alpha } _ { 1 }$ , and meanwhile the false positive will has a small margin $\hat { \alpha } _ { 2 }$ .
+
+# 3 The Proposed Method
+
+In this section, we elaborate on the proposed method, i.e., Noisy Correspondence Rectifier (NCR) which could be the first work to solve the noisy correspondence problem in cross-modal matching. In Section 3.1, we introduce the co-divide module which splits the training data into the clean and noisy subsets. After that, we introduce how to rectify the labels with an adaptive prediction function in Section 3.2. Finally, we detail how to combine the co-divide and co-rectify modules to achieve robust cross-modal matching in Section 3.3.
+
+# 3.1 Co-divide
+
+Without loss of generality, we first introduce the cross-modal matching task by taking the image-text matching as a showcase. Given the training data $\mathcal { D } = \{ ( I _ { i } , T _ { i } , y _ { i } ) \} _ { i = 1 } ^ { \overline { { N } } }$ , where $N$ is the data size, $( I _ { i } , T _ { i } )$ is an image-text pair and $y _ { i } \in \{ 1 , \bar { 0 } \}$ indicates that the pair belong to the same instance (positive) or not (negative). For the noisy correspondence case, it defines that an unknown portion of $\mathcal { D }$ is mismatched, i.e., $( I _ { i } , T _ { i } )$ is a negative pair but wrongly labeled as $y _ { i } = 1$ . To solve such a noisy correspondence problem, we propose NCR to achieve robust cross-modal matching.
+
+To begin, we project the visual and textual modalities into a shared space via two modal-specific networks $f$ and $g$ , respectively. Then the similarity of the given image-text pairs is computed through $S ( f ( I ) , g ( T ) )$ . For simplicity, we denote $S ( f ( \dot { I } ) , g ( T ) )$ as $S ( I , T )$ in the following. Some early empirical studies [3] show that DNNs tend to first learn simple samples and then gradually fit the noisy samples. This so-called memorization effect of DNNs will lead to a relatively low loss for the clean samples. Motivated by this, we utilize the difference of loss distribution between the clean samples and noisy samples to divide the training data like [10, 43, 2, 20]. Specifically, given a matching model $( \bar { f } , g , \bar { S } )$ , we compute the per-sample loss through:
+
+$$
+\ell _ { ( f , g , S ) } = \{ \ell _ { i } \} _ { i = 1 } ^ { N } = \{ L _ { w } ( I _ { i } , T _ { i } ) \} _ { i = 1 } ^ { N }
+$$
+
+where $L _ { w }$ is defined as:
+
+$$
+L _ { w } ( I , T ) = \sum _ { \hat { T } } [ \alpha - S ( I , T ) + S ( I , \hat { T } ) ] _ { + } + \sum _ { \hat { I } } [ \alpha - S ( I , T ) + S ( \hat { I } , T ) ] _ { + } ,
+$$
+
+where $( I , T )$ is a positive pair, $\alpha > 0$ denotes a given margin, and $[ x ] _ { + } = m a x ( x , 0 )$ . In the loss, the first term treats $I$ as queries taking over all negative text $\hat { T }$ , while the second term treats $T$ as queries taking over all negative images $\hat { I }$ . Then, we fit the per-sample loss of all training data by using a two-component Gaussian Mixture Model [20, 30]:
+
+$$
+p ( \ell | \theta ) = \sum _ { k = 1 } ^ { K } \beta _ { k } \phi ( \ell | k ) ,
+$$
+
+where $\beta _ { k }$ and $\phi ( \ell | k )$ are the mixture coefficient and the probability density of the $k$ -th component, respectively. Based on the memorization effect of DNNs, we treat the component with a lower mean value (i.e., lower loss) as the clean set, and the other as the noisy set. To optimize the GMM, we adopt the Expectation-Maximization algorithm. Moreover, we compute the posterior probability $w _ { i } \stackrel { - } { = } p ( k | \ell _ { i } ) ^ { - } = p ( k ) p ( \ell _ { i } | k ) / p ( \ell _ { i } )$ as the clean probability of $i$ -th sample, where $k$ is the Gaussian component with the lower mean. By setting a threshold to $\{ w _ { i } \} _ { i = 1 } ^ { N }$ , we divide the data into clean and noisy subsets. For simplicity, we set the threshold to 0.5 through all experiments.
+
+As observed in [10], it probably introduces error accumulation if the neural network is trained in a self-divide manner. To avoid such a situation, we adopt the co-teaching paradigm. Specifically, we individually train two networks $A = \{ f ^ { A } , g ^ { A } , S ^ { A } \}$ and $B = \{ f ^ { B } , \bar { g } ^ { B } , S ^ { \bar { B } } \}$ with different initializations and batch sequences. At each epoch, the network $A$ or $B$ will model its per-sample loss distribution with a GMM and divide the dataset into clean and noisy subsets which are then used for training the other network, i.e., co-divide. Note that, before co-divide, a warmup process is conducted on all training data to achieve initial convergence with $L _ { w }$ as defined in Eq. 2.
+
+# 3.2 Co-Rectify
+
+For either of A and B, the data $\mathcal { D }$ will be divided into the clean subset $\boldsymbol { \mathcal { S } } _ { c } = \{ ( \boldsymbol { I } _ { i } ^ { c } , \boldsymbol { T } _ { i } ^ { c } , y _ { i } ^ { c } , w _ { i } ) \} _ { i = 1 } ^ { N _ { c } }$ and noisy subset $\boldsymbol { \mathcal { S } } _ { n } = \{ I _ { i } ^ { n } , T _ { i } ^ { n } \} _ { i = 1 } ^ { N _ { n } }$ . Then, the co-rectify module will correct the labels to recall the possible true positives from $S _ { n }$ and eliminate the negative impact of the possible false positives in . Formally, the network $k$ $( k \in \{ A , B \} )$ will rectify the labels of $\{ S _ { c } , S _ { n } \}$ into $\{ \hat { \cal S } _ { c } , \hat { \cal S } _ { n } \}$ for training itself. The rectified labels are determined by:
+
+$$
+\left\{ \begin{array} { l r } { \hat { y } _ { i } ^ { c } = w _ { i } y _ { i } ^ { c } + ( 1 - w _ { i } ) P ^ { k } ( I _ { i } ^ { c } , T _ { i } ^ { c } ) , \qquad } & { \forall ( I _ { i } ^ { c } , T _ { i } ^ { c } , y _ { i } ^ { c } , w _ { i } ^ { c } ) \in \mathcal { S } _ { c } } \\ { \hat { y } _ { i } ^ { n } = ( P ^ { A } ( I _ { i } ^ { n } , T _ { i } ^ { n } ) + P ^ { B } ( I _ { i } ^ { n } , T _ { i } ^ { n } ) ) / 2 , } & { \forall ( I _ { i } ^ { n } , T _ { i } ^ { n } ) \in \mathcal { S } _ { n } } \end{array} \right.
+$$
+
+where $P ^ { A } ( I , T ) / P ^ { B } ( I , T )$ denotes the predictions given by the network $A / B$ . The roles of Eq. 4 are as below. On the one hand, as most pairs of $ { \boldsymbol { S } } _ { c }$ are true positive, Eq. 4 will use the original labels $y _ { i } ^ { c }$ together with the model’s prediction $P ( I _ { i } ^ { c } , T _ { i } ^ { c } )$ to rectify the correspondence. On the other hand, as most pairs of $S _ { n }$ are false positive, Eq. 4 will discard the original labels and rectify the labels by averaging the predictions $P ( I _ { i } ^ { n } , T _ { i } ^ { n } )$ from the networks A and B.
+
+Another key contribution of Eq. 4 is designing the prediction function $P ( I , T )$ that could accurately predict whether the given pairs are positive or negative. Unlike the tasks like classification, image-text matching aims at computing the similarity rather than predicting the label of given image-text pairs. To this end, a straightforward approach is to predict the pairs by setting a threshold on the similarity. However, such a method requires to specify the threshold value, which is a daunting task because the optimal value is actually the similarity boundary of positive and negative pairs and hard to be manually specified. Alternatively, the following adaptive prediction function $P ( I , T )$ is proposed, which could work in a data-driven way,
+
+$$
+\begin{array} { l } { \displaystyle P ( \boldsymbol { I } , \boldsymbol { T } ) = \Theta ( s ) / \tau } \\ { \displaystyle s = S ( I , T ) - ( \frac { 1 } { b } \sum _ { \hat { T } } S ( I , \hat { T } ) + \frac { 1 } { b } \sum _ { \hat { I } } S ( \hat { I } , T ) ) / 2 , } \end{array}
+$$
+
+where $b$ is the batch size, $\Theta ( \cdot )$ clamps the elements into the range of $[ 0 , \alpha ]$ , $s$ is the similarity margin between the given pair $( I , T )$ to the mean of the negatives in a mini-batch, $\tau$ is the average similarity
+
+margin of the largest $1 0 \%$ pairs in terms of $s$ . This implies that the data have at least $1 0 \%$ clean pairs which could be regarded as a similarity anchor for prediction. Intuitively, the pairs with the similarity margin larger than $\tau$ would be predicted as 1, otherwise $[ 0 , 1 )$ .
+
+# Algorithm 1: Noisy Correspondence Rectifier
+
+<table><tr><td>Input: A given training data D, matching models A = (fA,gA, SA) and B = (fB,gB,SB)</td></tr><tr><td>Warmup the model (A,B) using Lw· 2 forn=1:num_epoch do</td></tr><tr><td></td></tr><tr><td>3 ←GMM(D,B)</td></tr><tr><td>4 ← GMM(D,A) 5 for k={A,Bj do</td></tr><tr><td>Sk={(Ii,Ti,yi,Wi)lwi ≥0.5,∀(Ii,Ti,yi,Wi) ∈(D,Wk)} 6</td></tr><tr><td>S={(Ii,Ti)|ui&lt;0.5,∀(Ii,Ti) ∈(D,W)} 7</td></tr><tr><td>8 for j=num_steps do</td></tr><tr><td>9 Sample a mini-batch (B,Bη) from(Sk, S𝑘);</td></tr><tr><td>Rectify the labels of (Bj, Bγ) into (Bg, B) using Eq. 4-5; 10</td></tr><tr><td>Train the network k on (B, Bη） by optimizing Lsoft. 11</td></tr></table>
+
+Result: Matching models $( A , B )$
+
+# 3.3 Robust Cross-modal Matching
+
+Exiting cross-modal matching methods can only handle the binary labels which are incompatible with the soft labels rectified by NCR. To achieve robust image-text matching, we propose a novel triplet loss $\scriptstyle { L _ { s o f t } }$ by recasting the rectified labels as the soft margin. Mathematically,
+
+$$
+L _ { s o f t } ( I _ { i } , T _ { i } ) = [ \hat { \alpha } _ { i } - S ( I _ { i } , T _ { i } ) + S ( I _ { i } , \hat { T } _ { h } ) ] _ { + } + [ \hat { \alpha } _ { i } - S ( I _ { i } , T _ { i } ) + S ( \hat { I } _ { h } , T _ { i } ) ] _ { + } ,
+$$
+
+where $\hat { I } _ { h } = \mathrm { a r g m a x } _ { I _ { j } \ne I _ { i } } S ( I _ { j } , T _ { i } )$ and $\hat { T } _ { h } = \mathrm { a r g m a x } _ { T _ { j } \neq T _ { i } } S ( I _ { i } , T _ { j } )$ are the most similar negatives in the mini-batch for a given positive pair $( I _ { i } , T _ { i } )$ similar to ${ \mathrm { V S E } } { + } { + }$ [8]. The soft margin $\hat { \alpha } _ { i }$ is adaptively determined by:
+
+$$
+\hat { \alpha } _ { i } = \frac { m ^ { \hat { y } _ { i } } - 1 } { m - 1 } \alpha ,
+$$
+
+where $m$ is the curve parameter, and $\hat { y } _ { i }$ is the rectified label. The above formulation is designed to achieve the following goal, i.e., $\hat { \alpha } _ { i }$ will be assigned a small value if $\hat { y } _ { i }$ is close to 0, and a large value otherwise. Thanks to Eq. 6–7, the similarity of the pair $( I , T )$ will be larger than that of the negatives by an adaptive margin $\hat { \alpha } _ { i }$ .
+
+Despite the adaptive margin, another major difference between $L _ { w }$ and $L _ { s o f t }$ is that $L _ { s o f t }$ will use the hard negatives which are the most similar negative pairs. Although the hard negatives are helpful in improving the performance, $L _ { w }$ cannot be beneficial from it due to the existence of noisy correspondence. Specifically, it is expected that only the similarity of the true positives is larger than that of the hard negatives. However, in the case of noisy correspondence, the similarity of the false positives will also be larger than that of the hard negatives, thus leading to the unavailability of the hard negatives for $L _ { w }$ during the co-divide stage. The detail of NCR is presented in Algorithm. 1.
+
+# 3.4 Discussions on Matching Loss
+
+To achieve robust cross-modal matching with the refined soft labels, we design a soft Triplet loss by recasting the labels into soft margins. Recently, there are some works have been proposed to handle the soft labels in the matching model. For example, Wray et al. [38] recast the soft similarity into binary labels by directly setting a threshold on the predicted similarity. Kim et al. [15] proposes a log-ratio matching loss with a regularization defined by the label distance ratio, which is computed by the continuous labels. Liu et al. [25] introduces the hubness problem in image-text matching and proposes to consider all samples in a mini-batch and weights them according to both local and global statistics. Wray et al. [38] recasts the soft similarity into binary labels by directly setting a threshold on the predicted similarity. Different from them, NCR proposes to recast the rectified soft labels into soft margins in the triplet loss by assigning large margins to the true positive pairs and small ones to the false positive pairs. As a result and more importantly, our loss is specifically designed to solve the noisy correspondence problem whereas existing ones are not.
+
+# 4 Experiment
+
+In this section, we carry out experiments to verify the effectiveness of NCR in robust image-text matching. In the experiments, we use three benchmark datasets including Flickr30K [42], MS-COCO [23], and Conceptual Captions [35]. Among them, Conceptual Captions is with real noisy correspondence from the wild, and Flickr30K and MS-COCO are with simulated noisy correspondence.
+
+# 4.1 Datasets and Performance Measurements
+
+Three datasets are used in our experiments. To be specific, Flickr30K contains 31,000 images collected from the Flickr website with five captions each. Following [19], we use 1,000 images for validation, 1,000 images for testing, and the rest for training. MS-COCO contains 123,287 images with five captions each. We follow the data partition in [19] which consists of 113,287 training images, 5,000 validation images, and 5,000 test images. As Flickr30K and MS-COCO are well annotated, we simulate the noisy correspondence by randomly shuffling the captions of training images for a specific percentage, denoted by noise ratio. Conceptual Captions is a large-scale data consisting of 3.3M images with a single caption each. As this data set is harvested from the Internet, about $3 \% \sim 2 0 \%$ correspondences are incorrect [35]. In our experiments, we use a subset of Conceptual Captions for evaluation, named CC152K. Specifically, we randomly select 150,000 samples from the training split for training, 1,000 samples from the validation split for validation, and 1,000 samples from the validation split for testing.
+
+Following [19], for all images, we take the Faster-RCNN [33] detector provided by [1] to extract the top 36 region proposals of which each is encoded as a 2048-dimensional feature. For evaluation, we take the recall at K $. ( \mathbb { R } ^ { \ @ \mathbb { K } ) }$ as the measurement. In short, ${ \mathrm { R @ K } }$ is the fraction of queries for which the correct item is retrieved in the closest K points to the query. In the experiments, we report $\mathbf { R } \ @ 1$ , $\mathbf { R } @ 5$ , and $\mathbb { R } \ @ 1 0$ for a comprehensive evaluation.
+
+# 4.2 Implementation Details
+
+NCR is a general framework which could enable almost all existing cross-modal matching methods robust against noisy correspondence. To verify the effectiveness of our framework, SGR [7] is chosen to guarantee the robustness because it is the state of the art in image-text matching. In brief, the image regions and words are projected into a shared embedding space through a full-connected layer $( i . e . , f )$ and a Bi-GRU [34] $( i . e . , g )$ , respectively. For the similarity function $S$ , it will compute the similarity between the given image and text by combining the local and global features with the help of a graph reasoning technique proposed in [18]. Due to the space limitation, we leave more details and results in the supplemental material.
+
+We train our network using the Adam optimizer [16] with the default parameters and a batch size of 128. For fair comparisons, the networks $f$ and $g$ are the same with SGR, i.e., the word embedding size is 300 and the joint embedding space size is 2048. In addition, we fix the margin $\alpha = 0 . 2$ and $m = 1 0$ for the soft margin through the experiments. At the inference stage, we average the similarities predicted by network $A$ and $B$ for the retrieval evaluation. To avoid overfitting, we choose the best checkpoint in terms of the sum of the recalls on the validation set.
+
+# 4.3 Comparisons with State of The Arts
+
+In this section, we conduct comparisons on the three datasets. The baselines include SCAN [19], VSRN [21], IMRAM [5], SGRAF, SGR and SAF [7]. For Flickr30K and MS-COCO, we report the results with three different noise ratios, i.e., $0 \%$ , $20 \%$ , and $50 \%$ . In addition, we also report the results of SGR on the clean Flickr30K and MS-COCO by discarding the noisy pairs, denoted by SGR-C. Clearly, SGR-C is a quite strong baseline since the used data does not contain noisy correspondence. We do not report the results of SGRAF and SAF on the clean datasets since our framework only extends SGR in this paper.
+
+When the noise rate is $0 \%$ , we directly refer to the results reported in the corresponding papers. For the noisy cases, we train the baseline models with the recommended settings three times and report the best result. Note that for SGR, we found it is very sensitive to the noisy correspondence, as shown in Table. 2. To obtain a desirable result, we experimentally employ a pre-training process to SGR (denoted by ${ \mathrm { S G R } } ^ { * }$ ), namely, training the model with the vanilla triplet loss without hard negatives and then following the standard pipeline of SGR.
+
+Table 1: Image-Text Retrieval on Flickr30K and MS-COCO 1K.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2"></td><td colspan="5">Flickr30K</td><td colspan="6">MS-COCO</td></tr><tr><td colspan="3">Image→Text</td><td colspan="3">Text→Image</td><td colspan="2">Image→Text</td><td colspan="3">Text→Image</td></tr><tr><td>Noise</td><td>Methods</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1 R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td rowspan="7">0%</td><td>SCAN</td><td>67.4</td><td>90.3</td><td>95.8</td><td>48.6</td><td>77.7</td><td>85.2</td><td>69.2 93.6</td><td>97.6</td><td>56.0</td><td>86.5</td><td>93.5</td></tr><tr><td>VSRN</td><td>71.3</td><td>90.6</td><td>96.0</td><td>54.7</td><td>81.8 88.2</td><td>76.2</td><td>94.8</td><td>98.2</td><td>62.8</td><td>89.7</td><td>95.1</td></tr><tr><td>IMRAM</td><td>74.1</td><td>93.0</td><td>96.6</td><td>53.9</td><td>79.4</td><td>87.2</td><td>76.7 95.6</td><td>98.5</td><td>61.7</td><td>89.1</td><td>95.0</td></tr><tr><td>SAF</td><td>73.7</td><td>93.3</td><td>96.3</td><td>56.1</td><td>81.5</td><td>88.0</td><td>76.1 95.4</td><td>98.3</td><td>61.8</td><td>89.4</td><td>95.3</td></tr><tr><td>SGR</td><td>75.2</td><td>93.3</td><td>96.6</td><td>56.2</td><td>81.0</td><td>86.5</td><td>78.0 95.8</td><td>98.2</td><td>61.4</td><td>89.3</td><td>95.4</td></tr><tr><td>SGRAF NCR</td><td>77.8</td><td>94.1</td><td>97.4</td><td>58.5</td><td>83.0</td><td>88.8</td><td>79.6 96.2</td><td>98.5</td><td>63.2</td><td>90.7</td><td>96.1</td></tr><tr><td></td><td>77.3</td><td>94.0</td><td>97.5</td><td>59.6</td><td>84.4</td><td>89.9</td><td>78.7 95.8</td><td>98.5</td><td>63.3</td><td>90.4</td><td>95.8</td></tr><tr><td rowspan="7">20%</td><td>SCAN</td><td>59.1</td><td>83.4</td><td>90.4</td><td>36.6</td><td>67.0</td><td>77.5 78.2</td><td>66.2 25.1</td><td>91.0 96.4</td><td>45.0</td><td>80.2</td><td>89.3</td></tr><tr><td>VSRN IMRAM</td><td>58.1 63.0</td><td>82.6</td><td>89.3</td><td>40.7</td><td>68.7</td><td></td><td>59.0</td><td>74.8</td><td>17.6</td><td>49.0</td><td>64.1</td></tr><tr><td>SAF</td><td>51.0</td><td>86.0</td><td>91.3</td><td>41.4</td><td>71.2</td><td>80.5 68.6</td><td>92.8</td><td>97.6</td><td>55.7</td><td>85.0</td><td>91.0</td></tr><tr><td>SGR*</td><td>62.8</td><td>79.3</td><td>88.0</td><td>38.3</td><td>66.5</td><td>76.2</td><td>67.3 92.5 91.7</td><td>96.6 96.2</td><td>53.4 52.9</td><td>84.5</td><td>92.4</td></tr><tr><td>SGR-C</td><td>72.8</td><td>86.2 90.8</td><td>92.2 95.4</td><td>44.4 56.4</td><td>72.3</td><td>80.4</td><td>67.8</td><td></td><td></td><td>83.5</td><td>90.1</td></tr><tr><td>NCR</td><td>75.0</td><td>93.9</td><td>97.5</td><td></td><td>82.1</td><td>88.6</td><td>75.4 95.2</td><td>97.9</td><td>60.1</td><td>88.5 89.3</td><td>94.8</td></tr><tr><td></td><td></td><td></td><td></td><td>58.3</td><td>83.0</td><td>89.0</td><td>77.7</td><td>95.5</td><td>98.2</td><td>62.5</td><td></td><td>95.3 21.0</td></tr><tr><td rowspan="7">50%</td><td>SCAN</td><td>27.7</td><td>57.6</td><td>68.8</td><td>16.2</td><td>39.3</td><td>49.8</td><td>40.8</td><td>73.5 84.9</td><td>5.4</td><td></td><td>15.1</td></tr><tr><td>VSRN</td><td>14.3</td><td>37.6</td><td>50.0</td><td>12.1</td><td>30.0</td><td>39.4</td><td>23.5 54.7</td><td>69.3</td><td>16.0</td><td>47.8</td><td>65.9</td></tr><tr><td>IMRAM</td><td>9.1</td><td>26.6</td><td>38.2</td><td>2.7</td><td>8.4</td><td>12.7</td><td>21.3 60.2</td><td>75.9</td><td>22.3</td><td>52.8</td><td>64.3</td></tr><tr><td>SAF</td><td>30.3</td><td>63.6</td><td>75.4</td><td>27.9</td><td>53.7</td><td>65.1</td><td>30.4 67.8</td><td>82.3</td><td>33.5</td><td>69.0</td><td>82.8</td></tr><tr><td>SGR*</td><td>36.9</td><td>68.1</td><td>80.2</td><td>29.3</td><td>56.2</td><td>67.0</td><td>60.6 87.4</td><td>93.6</td><td>46.0</td><td>74.2</td><td>79.0</td></tr><tr><td>SGR-C</td><td>69.8</td><td>90.3</td><td>94.8</td><td>50.1</td><td>77.5</td><td>85.2</td><td>71.7 94.1</td><td>97.7</td><td>57.0</td><td>86.6</td><td>93.7</td></tr><tr><td>NCR</td><td>72.9</td><td>93.0</td><td>96.3</td><td>54.3</td><td>79.8</td><td>86.5</td><td>74.6</td><td>94.6</td><td>97.8 59.1</td><td>87.8</td><td>94.5</td></tr></table>
+
+Results on Flickr30K & MS-COCO. Table 1 shows the quantitative results on Flickr30K and MS-COCO. Note that for MS-COCO, we only report the results by averaging over 5 folds of 1K test images due to space limitation, and leave the results on the full 5K test images in the supplemental material. From the results, one could observe that NCR is competitive to SGRAF in the noisefree case, namely, NCR could achieve state-of-the-art performance even though it is proposed to achieve robustness. When the data is contaminated by the noisy correspondence, NCR remarkably outperforms all the baselines by a large margin. Even comparing with SGR-C which is trained on the clean data, NCR improves $\mathbb { R } \ @ 1$ by $2 . 2 \%$ , $3 . 1 \%$ , $2 . 3 \%$ , and $2 . 9 \%$ in these four valuations.
+
+Results on CC152K. Table 2 shows the quantitative results on the CC152K which is with real noisy correspondences. From the results, one could see that our NCR consistently outperforms the evaluated models by a considerable margin in terms of all metrics. Specifically, NCR is $4 . 5 \%$ and $5 . 4 \%$ higher than the best baseline in terms of $\mathbf { R } \ @ 1$ in text and image retrieval, respectively. Moreover, the large performance gap between SGR and $\operatorname { S G R } ^ { * }$ shows the noise sensitivity of the original SGR.
+
+Table 2: Image-Text Retrieval on CC152K.   
+
+<table><tr><td></td><td colspan="3">Image→ Text</td><td colspan="3">Text→Image</td></tr><tr><td>Methods</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td>SCAN (ECCV&#x27;18)</td><td>30.5</td><td>55.3</td><td>65.3</td><td>26.9</td><td>53.0</td><td>64.7</td></tr><tr><td>VSRN (ICCV&#x27;19)</td><td>32.6</td><td>61.3</td><td>70.5</td><td>32.5</td><td>59.4</td><td>70.4</td></tr><tr><td>IMRAM (CVPR&#x27;20)</td><td>33.1</td><td>57.6</td><td>68.1</td><td>29.0</td><td>56.8</td><td>67.4</td></tr><tr><td>SAF (AAAI&#x27;21)</td><td>31.7</td><td>59.3</td><td>68.2</td><td>31.9</td><td>59.0</td><td>67.9</td></tr><tr><td>SGR (AAAI&#x27;21)</td><td>11.3</td><td>29.7</td><td>39.6</td><td>13.1</td><td>30.1</td><td>41.6</td></tr><tr><td>SGR* (AAAI&#x27;21)</td><td>35.0</td><td>63.4</td><td>73.3</td><td>34.9</td><td>63.0</td><td>72.8</td></tr><tr><td>NCR</td><td>39.5</td><td>64.5</td><td>73.5</td><td>40.3</td><td>64.6</td><td>73.2</td></tr></table>
+
+# 4.4 Comparison to pre-trained model
+
+In this section, we perform comparison to the large pre-trained model CLIP [31]. In brief, CLIP is trained on a massive dataset harvested from the Internet and thus presumably has a lot of noisy image-text pairs. Such a comparison is helpful in understanding, big data based model (CLIP) or noisy correspondence modeling technique (NCR), which one is more favorable to handle the mismatching problem. More specifically, CLIP claims that using hundreds of million data could ignore the existence of possible noise, while we believe that a well-designed algorithm is essential to solve the noisy correspondence. Noticed, although some existing works including CLIP have slightly/indirectly realized the existence of the noise, NONE of them explicitly give a solution to solve this problem and explores the characteristics of the noise correspondence.
+
+In the experiments, we conduct CLIP on the MS-COCO dataset under the following two settings, i.e., Zero-shot and Fine-tune. In brief, the first setting directly uses the released pre-trained CLIP to perform inference on MS-COCO, and the second fine-tunes the pre-trained model using the noisy training data of MS-COCO. As CLIP only released some pre-trained models and inference code 3, we use the non-official code 4 to fine-tune the model with 32 epochs for the fine-tune setting. Note that CLIP (ViT-L/14†) is unreleased and we report the results from the original paper [31]. One could observe that, although CLIP utilizes 400 million image-text pairs for pre-training, its performance inevitably degenerates during fine-tuning. In contrast, NCR achieves the matching performance with the presence of noisy correspondence, indicating the necessity of algorithm design.
+
+Table 3: Comparison with CLIP on MS-COCO 5K.   
+
+<table><tr><td></td><td></td><td colspan="3">Image→ Text</td><td colspan="3">Text→Image</td></tr><tr><td>Noise Ratio</td><td>Methods</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td rowspan="2">0%, Zero-Shot</td><td>CLIP (ViT-L/14+)</td><td>58.4</td><td>81.5</td><td>88.1</td><td>37.8</td><td>62.4</td><td>72.2</td></tr><tr><td>CLIP (ViT-B/32)</td><td>50.2</td><td>74.6</td><td>83.6</td><td>30.4</td><td>56.0</td><td>66.8</td></tr><tr><td rowspan="2">20%,Fine-tune</td><td>NCR</td><td>58.2</td><td>84.2</td><td>91.5</td><td>41.7</td><td>71.0</td><td>81.3</td></tr><tr><td>CLIP (ViT-B/32) NCR</td><td>21.4 56.9</td><td>49.6 83.6</td><td>63.3 91.0</td><td>14.8 40.6</td><td>37.6 69.8</td><td>49.6 80.1</td></tr><tr><td rowspan="2">50%, Fine-tune</td><td>CLIP (ViT-B/32)</td><td>10.9</td><td>27.8</td><td>38.3</td><td>7.8</td><td>19.5</td><td></td></tr><tr><td>NCR</td><td>53.1</td><td>80.7</td><td>88.5</td><td>37.9</td><td>66.6</td><td>26.8 77.8</td></tr></table>
+
+# 4.5 Experimental Analysis
+
+In this section, we first conduct experiments to show the robustness and generalizability of the proposed method. Then, we investigate the effect of co-divide and co-rectify with the visualization results. After that, we carry out the ablation study to verify different components of NCR. Finally, we visually demonstrate some noisy cases detected by NCR.
+
+# 4.5.1 Study on Robustness and Generalizability
+
+To show the robustness of NCR, we conduct experiments on Flickr30K by increasing the noise ratio from $0 \%$ to $6 0 \%$ with an interval of $1 0 \%$ . In addition, to verify the generalizability of NCR to other image-text matching methods, we extend SCAN [19] by NCR, denoted by NCR-SCAN. Fig. 4 shows that NCR and NCR-SCAN perform more stable than SGR and SCAN with increasing noise ratio. Moreover, NCR (NCR-SCAN) is remarkably superior to SGR (SCAN) in all tests, which shows the generalizability of NCR.
+
+# 4.5.2 Visualization on the Co-divide and Co-rectify
+
+To further investigate the influence of the co-divide and co-rectify modules in our method, we carry out experiments on the Flickr30K dataset by visualizing the per-sample loss distribution and the model predictions on the noisy data, where the noisy ratio is $20 \%$ . For better visualization, here we show the result of NCR-SCAN and leave the result of NCR in the supplemental material. As shown in Fig. 3(b), the loss of most noisy samples is larger than the clean loss, which verifies the memorization effect of DNNs. By fitting the per-sample loss with GMM, NCR could effectively divide the data into clean and noisy splits. Regarding the analysis on the co-rectify, Fig. 3(c) shows that the rectified soft labels of most clean pairs range into [0.3, 1] and those of most noisy pairs range into [0, 0.5]. In other words, one could enforce the similarity of true positives larger than that of the negatives during training, thus eliminating the negative impact of the noisy correspondence.
+
+![](images/0503d947414e2b32158fb766cbf1a3b87cabce246d93f4abb1f73ad11d4aa0c3.jpg)  
+Figure 3: (a) Retrieval performance of NCR and NCR-SCAN on Flickr30K with varying noise ratio. (b) Per-sample loss distribution and GMM fitting visualization after warmup. (c) Model predictions on the noisy subset.
+
+# 4.5.3 Ablation Study
+
+In this section, we carry out the ablation study on the Flickr30K with the noise ratio of $50 \%$ . As shown in Table 4, all these three components are important to achieving encouraging results.
+
+Table 4: Ablation studies on Flickr30K with $50 \%$ noise.   
+
+<table><tr><td colspan="3">Method</td><td colspan="3">Image →Text</td><td colspan="3">Text→Image</td></tr><tr><td>Co-divide</td><td>Co-rectify</td><td>Warmup</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td>?</td><td>√</td><td>&lt;&lt;&gt;</td><td>72.9</td><td>93.0</td><td>96.3</td><td>54.3</td><td>79.8</td><td>86.5</td></tr><tr><td></td><td></td><td></td><td>71.4</td><td>90.8</td><td>95.7</td><td>54.1</td><td>80.3</td><td>86.5</td></tr><tr><td></td><td>?</td><td></td><td>16.0</td><td>38.4</td><td>51.7</td><td>12.6</td><td>31.4</td><td>42.8</td></tr><tr><td>√</td><td></td><td></td><td>0.3</td><td>0.6</td><td>1.0</td><td>0.2</td><td>0.5</td><td>1.1</td></tr></table>
+
+# 4.5.4 Noisy Samples
+
+Fig. 4 shows some noisy CC152K examples identified by NCR. As shown, the first four image-text pairs are completely unrelated, which will be successfully detected by NCR. For the last pair, it will also be detected as noisy correspondence even though the visual and textual modalities are correlated at a coarse-grained level, e.g., both the image and text involve “beach".
+
+![](images/c2ea469f89d768984f1c578d0e0317f6d920992f64f93e122eb3703e75d9c2da.jpg)  
+Figure 4: Some noisy examples correctly divided by NCR.
+
+# 5 Conclusion
+
+This paper could be the first attempt to study a new problem in cross-modal matching, i.e., the noisy correspondence which could be a potential new direction in noise label. To solve this problem in cross-modal matching, we propose rectifying the noisy correspondence by an adaptive prediction function and a novel triplet loss with a soft margin to achieve robust cross-modal matching. Extensive experiments verify the effectiveness of the proposed method in handling synthesized and real noisy correspondences.
+
+# Broader Impact Statement
+
+Cross-modal matching is a fundamental topic in multimodal learning, which could be applied to a wide range of applications including data retrieval, recommender systems, and vision-and-language understanding. This work could be one of the first works to aware of the importance and existence of the noisy correspondence problem in numerous applications. There are many benefits to solving the noisy correspondence problem, e.g., reducing the costs for manually annotating and aligning data; more data could be collected and used even though some of them are incorrectly aligned. Besides the benefits, it should pay attention on the potential negative impacts including but not limited to 1) The risk of automation bias [28] for decision making from the data bias, especially in aviation, health care, and autonomous vehicles. 2) The job loss caused by the NCR since it makes possibility to automatically correct the noisy correspondence, thus remarkably reducing the cost of human labor. We would encourage further work to understand and mitigate the above biases and risks.
+
+# Acknowledgements
+
+The authors would thank to the anonymous reviewers whose valuable suggestions and constructive comments remarkably improve this work. This work was supported in part by the National Key R&D Program of China under Grant 2020AAA0104500; in part by the Key Research and Development Program of Sichuan Province under Grant 2019YFG0497; in part by NFSC under Grant U21B2040, 62176171, U19A2078, 61625204, and 61836006; in part by the Fund of Sichuan University Tomorrow Advancing Life.
+
+# References
+
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+        "text": "Zhenyu Huang∗ College of Computer Science Sichuan University, China zyhuang.gm@gmail.com ",
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+        "type": "text",
+        "text": "Xi Peng† College of Computer Science Sichuan University, China pengx.gm@gmail.com ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Cross-modal matching, which aims to establish the correspondence between two different modalities, is fundamental to a variety of tasks such as cross-modal retrieval and vision-and-language understanding. Although a huge number of crossmodal matching methods have been proposed and achieved remarkable progress in recent years, almost all of these methods implicitly assume that the multimodal training data are correctly aligned. In practice, however, such an assumption is extremely expensive even impossible to satisfy. Based on this observation, we reveal and study a latent and challenging direction in cross-modal matching, named noisy correspondence, which could be regarded as a new paradigm of noisy labels. Different from the traditional noisy labels which mainly refer to the errors in category labels, our noisy correspondence refers to the mismatch paired samples. To solve this new problem, we propose a novel method for learning with noisy correspondence, named Noisy Correspondence Rectifier (NCR). In brief, NCR divides the data into clean and noisy partitions based on the memorization effect of neural networks and then rectifies the correspondence via an adaptive prediction model in a co-teaching manner. To verify the effectiveness of our method, we conduct experiments by using the image-text matching as a showcase. Extensive experiments on Flickr30K, MS-COCO, and Conceptual Captions verify the effectiveness of our method. The code could be accessed from www.pengxi. me. ",
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+        "text": "1 Introduction ",
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+        "text": "As one of the most fundamental techniques in multimodal learning, cross-modal matching aims to bridge different modalities. In recent years, some cross-modal matching methods [19, 11, 7, 26] have been proposed based on Deep Neural Networks (DNNs), which achieved remarkable progress in a variety of applications, such as clustering [29, 24], image/video captioning [1, 44, 22], cross-modal retrieval [40, 19, 13], and visual question answering [9]. ",
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+        "text": "In general, most existing cross-modal matching methods embed different modalities into a common space wherein the similarity of positive cross-modal pairs is maximized and that of the negative ones is minimized. Although these methods have achieved promising results, their success depends on an implicit data assumption, i.e., the training data are correctly aligned across modalities. For example, in the vision-and-language tasks, the text needs to accurately describe the image content, and vice versa. In practice, however, it is extremely expensive and time-consuming to annotate or collect such data pairs. Especially, considering the data collected from the Internet [35, 14], it is inevitable to collect some mismatched pairs which are wrongly treated as the matched ones. To the best of our knowledge, such a special noisy label (correspondence) problem has been ignored so far, which will remarkably degrade the performance of matching methods as shown in our experiments. ",
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+        "image_caption": [
+            "Figure 1: Noisy labels vs. Noisy Correspondence. We denote the noisy samples with red lines and clean samples with green lines. The traditional noisy labels mainly refer to the errors in category labels, while the noisy correspondence refers to the alignment errors in paired data. For the noisy correspondence in cross-modal matching, the true positive pair correctly guides the cross-modal matching, while the false positive pair causes incorrect supervision for training. "
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+        "text": "Based on the above observation, we reveal a new paradigm for the noisy labels, named noisy correspondence. Different from the traditional noisy labels, the noisy correspondence refers to the alignment errors in paired data rather than the errors in category annotations (see Fig. 1). To the best of our knowledge, there is no effort has been devoted to study this new problem and the closest paradigm might be the partially view-aligned problem (PVP) [12, 41]. However, PVP is remarkably different from noisy correspondence, and the latter is more practical than the former. To be specific, PVP focuses on that the cross-modal alignment is totally unavailable, whereas the noisy correspondence focuses on that some correspondences are incorrect. In addition, PVP assumes that some correctly aligned data are available for training, whereas our noisy correspondence assumes that the clean and noisy data are mixed. ",
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+    {
+        "type": "text",
+        "text": "To solve the noisy correspondence problem in cross-modal matching, we propose a novel method, named Noisy Correspondence Rectifier (NCR). Our method is based on the memorization effect of DNNs observed in [3, 39], i.e., DNNs tend to learn the simple patterns before fitting noisy samples. Motivated by this empirical observation, NCR divides the data into two relative accurate data partitions, i.e., “noisy” and “clean” subsets, based on their loss difference. After that, NCR employs an adaptive prediction function for label rectifying so that the false positives and the true positives could be identified from the “clean” and the “noisy” subsets, respectively. Furthermore, we propose a novel triplet loss for robust cross-modal matching by recasting the rectified labels as the soft margin. ",
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+    {
+        "type": "text",
+        "text": "The main contributions and novelties of this paper could be summarized as below. i) We reveal a new problem in cross-modal analysis, which is also a new paradigm for noisy labels, termed noisy correspondence. Different from the traditional noisy labels, the noisy correspondence refers to the alignment errors in paired data instead of the errors in category annotations. To the best of our knowledge, this work could be the first study on this problem. ii) To solve the noisy correspondence problem, we propose a novel method for learning with noisy correspondence, named Noisy Correspondence Rectifier (NCR). One major novelty of NCR is that the rectified label is elegantly recasted as the soft margin of a triplet loss so that the robust cross-modal matching could be achieved. iii) To verify the effectiveness of our method, we conduct experiments on the image-text matching task. Extensive experiments on three challenging datasets verify the effectiveness of our method in the synthesized and real noises. ",
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+        "text": "2 Related works ",
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+    {
+        "type": "text",
+        "text": "In this section, we briefly introduce some recent developments in cross-modal matching and learning with noisy labels. ",
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+        "text": "2.1 Cross-modal Matching ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Most existing cross-modal matching works seek to learn a common space wherein different modalities are comparable. In general, existing works could be roughly divided into two categories: 1) Coarsegrained Matching. It often utilizes multiple neural networks to compute a global feature and each network is used for a specific modality [17, 37, 8]. For example, Kiros et al. [17] use a Convolutional Neural Network (CNN) and a Gated Recurrent Unit (GRU) to obtain the image and text features, while enforcing the similarity of positive pairs larger than that of the negative ones. To further boost the matching performance, ${ \\mathrm { V S E } } { + } { + }$ [8] uses some representative negatives to improve the discrimination of the model. 2) Fine-grained Matching. It seeks to measure the fine-grained similarity for cross-modal matching [19, 21, 7]. For example, SCAN [19] proposes learning the latent semantic correspondence between the image regions and words that are extracted by bottom-up attention [1] and GRU, respectively. VSRN [21] adopts a graph convolutional network for semantic reasoning. SGRAF [7] proposes constructing a similarity graph to reason the similarity and adopting an attention filtration technique to eliminate the less-meaningful alignments. Recently Chun et al. [6] introduce a new paradigm for cross-modal matching, i.e. possible many-to-many correspondence that existed in the image and captions. To achieve this, they propose to use probabilistic representations to model the possible one-to-many correspondence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although promising results have been achieved in recent years, the existing methods heavily rely on the correctly aligned data. In practice, however, such well-matched data is expensive and timeconsuming to collect. Moreover, some recent works [35, 14] show that a large-scale dataset collected from the wild could remarkably improve the performance of the model. However, such a data will inevitably contain some mismatched pairs. Hence, it is highly expected to develop some methods which are robust against the noisy correspondence, which has not been studied as far as we know. Different from the many-to-many correspondence [6] between image and captions, NCR reveals the noisy correspondence problem which refers to the alignment errors of image-text pairs and proposes to eliminate the negative impact from noisy pairs for downstream tasks. ",
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+        "type": "text",
+        "text": "2.2 Learning with Noisy Labels ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To handle the possible noisy annotations in the training data, a large number of methods have been proposed and almost all of them focus on the classification task [36, 27]. To reduce the negative impact of the noisy labels, the existing works often resort to robust architecture design, regularization, loss adjustment, or sample selection methods. Here, we mainly introduce the last two approaches which are most related to this work. To be specific, the loss adjustment achieves robustness by adjusting the contribution of clean and noisy samples w.r.t. the loss. For example, Reed et al. [32] proposed a bootstrapping loss based on the model predictions for loss correction. Zhang et al. [45] provided some theoretical explanations for the label correction along with a new label correction algorithm. Different from the loss adjustment methods, sample selection methods aim to select clean samples from a noisy dataset. For example, Arpit et al. [3] showed that DNNs tend to learn simple patterns before fitting noisy samples, namely the memorization effect. Motivated from this, Arazo et al. [2] proposed treating the samples with small loss as the clean samples. To avoid the selection bias of clean samples, Co-teaching methods [10, 43] use the samples with small loss to iteratively train two networks. In recent, DivideMix [20] adopts the MixMatch method [4] for semi-supervised learning with the clean and noisy samples. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Unlike the above noisy label studies, this paper focuses on the noisy correspondence problem which considers mismatched multimodal data pairs instead of incorrectly annotated data points. Besides the difference in the problem, this work is also different from the aforementioned studies in the methodology. To be specific, in cross-modal matching, it is impossible to directly adopt these noise label learning methods to solve the noisy correspondence problem due to the following two reasons. First, most of the noisy label learning methods propose to use the model’s prediction for label rectifying in the scenario of classification, while it is intractable to directly predict the correspondence of given pairs in matching models. Second, even if we can rectify the noisy correspondence somehow, the rectified real-valued labels are incompatible with the existing matching methods since most of them assume the given labels are binary. To address these problems, NCR proposes an adaptive prediction function and a novel triplet loss by recasting the soft labels as soft margins. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5c322d44fac00fe31e4ff1c8ce4a4a36193b7016a6b16a32f08f9c75ab7a153c.jpg",
+        "image_caption": [
+            "Figure 2: Overview of the proposed method. (a) Training pipeline of NCR. NCR consists of two individual networks (A, B) which work in the manner of co-teaching. In brief, NCR first warmup the networks (A, B) on the original training data using the loss $L _ { w }$ which is also used for per-sample loss computation. Then, based on the memorization effect of DNclean and noisy subsets at each epoch using either A or B, i.e., $S ^ { A } = ( S _ { c } ^ { A } , S _ { n } ^ { A } )$ the and $S ^ { B } = \\mathsf { \\bar { ( } } S _ { c } ^ { B } , S _ { n } ^ { B } )$ After that, NCR will co-rectify the correspondence of $\\{ { \\cal S } ^ { A } , { \\cal S } ^ { B } \\}$ and obtain $\\{ \\hat { S ^ { A } } , \\hat { S ^ { B } } \\}$ using an adaptive prediction function. Finally, $\\hat { \\cal S } ^ { A }$ and $\\hat { S ^ { B } }$ will be used to train the network $B$ and $A$ in a swapping way. (b) Robust image-text matching network. For example, the network A projects the image and text by the modal-specific networks $f$ and $g$ , respectively. Then the similarity $S ( I , T )$ is computed on the extracted features $f ( I )$ and $g ( T )$ . To achieve robust image-text matching, the rectified soft labels are recast as the soft margin of our loss $\\scriptstyle L _ { s o f t }$ . As shown, for a given anchor, $\\boldsymbol { L _ { s o f t } }$ will enforce the true positive to closer to it than the negative by a large margin $\\hat { \\alpha } _ { 1 }$ , and meanwhile the false positive will has a small margin $\\hat { \\alpha } _ { 2 }$ . "
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+        "type": "text",
+        "text": "3 The Proposed Method ",
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+        "text": "In this section, we elaborate on the proposed method, i.e., Noisy Correspondence Rectifier (NCR) which could be the first work to solve the noisy correspondence problem in cross-modal matching. In Section 3.1, we introduce the co-divide module which splits the training data into the clean and noisy subsets. After that, we introduce how to rectify the labels with an adaptive prediction function in Section 3.2. Finally, we detail how to combine the co-divide and co-rectify modules to achieve robust cross-modal matching in Section 3.3. ",
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+        "type": "text",
+        "text": "3.1 Co-divide ",
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+        "text": "Without loss of generality, we first introduce the cross-modal matching task by taking the image-text matching as a showcase. Given the training data $\\mathcal { D } = \\{ ( I _ { i } , T _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { \\overline { { N } } }$ , where $N$ is the data size, $( I _ { i } , T _ { i } )$ is an image-text pair and $y _ { i } \\in \\{ 1 , \\bar { 0 } \\}$ indicates that the pair belong to the same instance (positive) or not (negative). For the noisy correspondence case, it defines that an unknown portion of $\\mathcal { D }$ is mismatched, i.e., $( I _ { i } , T _ { i } )$ is a negative pair but wrongly labeled as $y _ { i } = 1$ . To solve such a noisy correspondence problem, we propose NCR to achieve robust cross-modal matching. ",
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+        "text": "To begin, we project the visual and textual modalities into a shared space via two modal-specific networks $f$ and $g$ , respectively. Then the similarity of the given image-text pairs is computed through $S ( f ( I ) , g ( T ) )$ . For simplicity, we denote $S ( f ( \\dot { I } ) , g ( T ) )$ as $S ( I , T )$ in the following. Some early empirical studies [3] show that DNNs tend to first learn simple samples and then gradually fit the noisy samples. This so-called memorization effect of DNNs will lead to a relatively low loss for the clean samples. Motivated by this, we utilize the difference of loss distribution between the clean samples and noisy samples to divide the training data like [10, 43, 2, 20]. Specifically, given a matching model $( \\bar { f } , g , \\bar { S } )$ , we compute the per-sample loss through: ",
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+        "img_path": "images/6fc5284745d0ad1346714b29253da56ab6a8e8fa66ab3747454a6042d2257c71.jpg",
+        "text": "$$\n\\ell _ { ( f , g , S ) } = \\{ \\ell _ { i } \\} _ { i = 1 } ^ { N } = \\{ L _ { w } ( I _ { i } , T _ { i } ) \\} _ { i = 1 } ^ { N }\n$$",
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+        "type": "text",
+        "text": "where $L _ { w }$ is defined as: ",
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+        "type": "equation",
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+        "text": "$$\nL _ { w } ( I , T ) = \\sum _ { \\hat { T } } [ \\alpha - S ( I , T ) + S ( I , \\hat { T } ) ] _ { + } + \\sum _ { \\hat { I } } [ \\alpha - S ( I , T ) + S ( \\hat { I } , T ) ] _ { + } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $( I , T )$ is a positive pair, $\\alpha > 0$ denotes a given margin, and $[ x ] _ { + } = m a x ( x , 0 )$ . In the loss, the first term treats $I$ as queries taking over all negative text $\\hat { T }$ , while the second term treats $T$ as queries taking over all negative images $\\hat { I }$ . Then, we fit the per-sample loss of all training data by using a two-component Gaussian Mixture Model [20, 30]: ",
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+        "type": "equation",
+        "img_path": "images/3b664f0344f0c21be2caace0bac7efb85763fbbc8bd31b5bf4a18615525f996b.jpg",
+        "text": "$$\np ( \\ell | \\theta ) = \\sum _ { k = 1 } ^ { K } \\beta _ { k } \\phi ( \\ell | k ) ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\beta _ { k }$ and $\\phi ( \\ell | k )$ are the mixture coefficient and the probability density of the $k$ -th component, respectively. Based on the memorization effect of DNNs, we treat the component with a lower mean value (i.e., lower loss) as the clean set, and the other as the noisy set. To optimize the GMM, we adopt the Expectation-Maximization algorithm. Moreover, we compute the posterior probability $w _ { i } \\stackrel { - } { = } p ( k | \\ell _ { i } ) ^ { - } = p ( k ) p ( \\ell _ { i } | k ) / p ( \\ell _ { i } )$ as the clean probability of $i$ -th sample, where $k$ is the Gaussian component with the lower mean. By setting a threshold to $\\{ w _ { i } \\} _ { i = 1 } ^ { N }$ , we divide the data into clean and noisy subsets. For simplicity, we set the threshold to 0.5 through all experiments. ",
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+        "type": "text",
+        "text": "As observed in [10], it probably introduces error accumulation if the neural network is trained in a self-divide manner. To avoid such a situation, we adopt the co-teaching paradigm. Specifically, we individually train two networks $A = \\{ f ^ { A } , g ^ { A } , S ^ { A } \\}$ and $B = \\{ f ^ { B } , \\bar { g } ^ { B } , S ^ { \\bar { B } } \\}$ with different initializations and batch sequences. At each epoch, the network $A$ or $B$ will model its per-sample loss distribution with a GMM and divide the dataset into clean and noisy subsets which are then used for training the other network, i.e., co-divide. Note that, before co-divide, a warmup process is conducted on all training data to achieve initial convergence with $L _ { w }$ as defined in Eq. 2. ",
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+        "type": "text",
+        "text": "3.2 Co-Rectify ",
+        "text_level": 1,
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+        "text": "For either of A and B, the data $\\mathcal { D }$ will be divided into the clean subset $\\boldsymbol { \\mathcal { S } } _ { c } = \\{ ( \\boldsymbol { I } _ { i } ^ { c } , \\boldsymbol { T } _ { i } ^ { c } , y _ { i } ^ { c } , w _ { i } ) \\} _ { i = 1 } ^ { N _ { c } }$ and noisy subset $\\boldsymbol { \\mathcal { S } } _ { n } = \\{ I _ { i } ^ { n } , T _ { i } ^ { n } \\} _ { i = 1 } ^ { N _ { n } }$ . Then, the co-rectify module will correct the labels to recall the possible true positives from $S _ { n }$ and eliminate the negative impact of the possible false positives in . Formally, the network $k$ $( k \\in \\{ A , B \\} )$ will rectify the labels of $\\{ S _ { c } , S _ { n } \\}$ into $\\{ \\hat { \\cal S } _ { c } , \\hat { \\cal S } _ { n } \\}$ for training itself. The rectified labels are determined by: ",
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+        "type": "equation",
+        "img_path": "images/4e3b5a31990709244ff49efc174e8e2f4deeb4362d006d1e7b51bcade2efc57a.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l r } { \\hat { y } _ { i } ^ { c } = w _ { i } y _ { i } ^ { c } + ( 1 - w _ { i } ) P ^ { k } ( I _ { i } ^ { c } , T _ { i } ^ { c } ) , \\qquad } & { \\forall ( I _ { i } ^ { c } , T _ { i } ^ { c } , y _ { i } ^ { c } , w _ { i } ^ { c } ) \\in \\mathcal { S } _ { c } } \\\\ { \\hat { y } _ { i } ^ { n } = ( P ^ { A } ( I _ { i } ^ { n } , T _ { i } ^ { n } ) + P ^ { B } ( I _ { i } ^ { n } , T _ { i } ^ { n } ) ) / 2 , } & { \\forall ( I _ { i } ^ { n } , T _ { i } ^ { n } ) \\in \\mathcal { S } _ { n } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $P ^ { A } ( I , T ) / P ^ { B } ( I , T )$ denotes the predictions given by the network $A / B$ . The roles of Eq. 4 are as below. On the one hand, as most pairs of $ { \\boldsymbol { S } } _ { c }$ are true positive, Eq. 4 will use the original labels $y _ { i } ^ { c }$ together with the model’s prediction $P ( I _ { i } ^ { c } , T _ { i } ^ { c } )$ to rectify the correspondence. On the other hand, as most pairs of $S _ { n }$ are false positive, Eq. 4 will discard the original labels and rectify the labels by averaging the predictions $P ( I _ { i } ^ { n } , T _ { i } ^ { n } )$ from the networks A and B. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Another key contribution of Eq. 4 is designing the prediction function $P ( I , T )$ that could accurately predict whether the given pairs are positive or negative. Unlike the tasks like classification, image-text matching aims at computing the similarity rather than predicting the label of given image-text pairs. To this end, a straightforward approach is to predict the pairs by setting a threshold on the similarity. However, such a method requires to specify the threshold value, which is a daunting task because the optimal value is actually the similarity boundary of positive and negative pairs and hard to be manually specified. Alternatively, the following adaptive prediction function $P ( I , T )$ is proposed, which could work in a data-driven way, ",
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+    {
+        "type": "equation",
+        "img_path": "images/cf5164f6601aaf7e50d5373e1f70a6f56e186d9be31bcbf054a80a164bbeb171.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle P ( \\boldsymbol { I } , \\boldsymbol { T } ) = \\Theta ( s ) / \\tau } \\\\ { \\displaystyle s = S ( I , T ) - ( \\frac { 1 } { b } \\sum _ { \\hat { T } } S ( I , \\hat { T } ) + \\frac { 1 } { b } \\sum _ { \\hat { I } } S ( \\hat { I } , T ) ) / 2 , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $b$ is the batch size, $\\Theta ( \\cdot )$ clamps the elements into the range of $[ 0 , \\alpha ]$ , $s$ is the similarity margin between the given pair $( I , T )$ to the mean of the negatives in a mini-batch, $\\tau$ is the average similarity ",
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+    {
+        "type": "text",
+        "text": "margin of the largest $1 0 \\%$ pairs in terms of $s$ . This implies that the data have at least $1 0 \\%$ clean pairs which could be regarded as a similarity anchor for prediction. Intuitively, the pairs with the similarity margin larger than $\\tau$ would be predicted as 1, otherwise $[ 0 , 1 )$ . ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Noisy Correspondence Rectifier ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/5594c7893fc3111e104dd235ec8d9a648eb8693d4b754bb39fa6af022f171953.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Result: Matching models $( A , B )$ "
+        ],
+        "table_body": "<table><tr><td>Input: A given training data D, matching models A = (fA,gA, SA) and B = (fB,gB,SB)</td></tr><tr><td>Warmup the model (A,B) using Lw· 2 forn=1:num_epoch do</td></tr><tr><td></td></tr><tr><td>3 ←GMM(D,B)</td></tr><tr><td>4 ← GMM(D,A) 5 for k={A,Bj do</td></tr><tr><td>Sk={(Ii,Ti,yi,Wi)lwi ≥0.5,∀(Ii,Ti,yi,Wi) ∈(D,Wk)} 6</td></tr><tr><td>S={(Ii,Ti)|ui&lt;0.5,∀(Ii,Ti) ∈(D,W)} 7</td></tr><tr><td>8 for j=num_steps do</td></tr><tr><td>9 Sample a mini-batch (B,Bη) from(Sk, S𝑘);</td></tr><tr><td>Rectify the labels of (Bj, Bγ) into (Bg, B) using Eq. 4-5; 10</td></tr><tr><td>Train the network k on (B, Bη） by optimizing Lsoft. 11</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.3 Robust Cross-modal Matching ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Exiting cross-modal matching methods can only handle the binary labels which are incompatible with the soft labels rectified by NCR. To achieve robust image-text matching, we propose a novel triplet loss $\\scriptstyle { L _ { s o f t } }$ by recasting the rectified labels as the soft margin. Mathematically, ",
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+        "type": "equation",
+        "img_path": "images/b89b0624abfbe280229b638d5dd7358f01553e7b11c483fbe264ed86470ec138.jpg",
+        "text": "$$\nL _ { s o f t } ( I _ { i } , T _ { i } ) = [ \\hat { \\alpha } _ { i } - S ( I _ { i } , T _ { i } ) + S ( I _ { i } , \\hat { T } _ { h } ) ] _ { + } + [ \\hat { \\alpha } _ { i } - S ( I _ { i } , T _ { i } ) + S ( \\hat { I } _ { h } , T _ { i } ) ] _ { + } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\hat { I } _ { h } = \\mathrm { a r g m a x } _ { I _ { j } \\ne I _ { i } } S ( I _ { j } , T _ { i } )$ and $\\hat { T } _ { h } = \\mathrm { a r g m a x } _ { T _ { j } \\neq T _ { i } } S ( I _ { i } , T _ { j } )$ are the most similar negatives in the mini-batch for a given positive pair $( I _ { i } , T _ { i } )$ similar to ${ \\mathrm { V S E } } { + } { + }$ [8]. The soft margin $\\hat { \\alpha } _ { i }$ is adaptively determined by: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/a876de5ed3ec8e6b239cb30ae499449cd6fc1e4190ca088b63be19780b097d3d.jpg",
+        "text": "$$\n\\hat { \\alpha } _ { i } = \\frac { m ^ { \\hat { y } _ { i } } - 1 } { m - 1 } \\alpha ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $m$ is the curve parameter, and $\\hat { y } _ { i }$ is the rectified label. The above formulation is designed to achieve the following goal, i.e., $\\hat { \\alpha } _ { i }$ will be assigned a small value if $\\hat { y } _ { i }$ is close to 0, and a large value otherwise. Thanks to Eq. 6–7, the similarity of the pair $( I , T )$ will be larger than that of the negatives by an adaptive margin $\\hat { \\alpha } _ { i }$ . ",
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+    {
+        "type": "text",
+        "text": "Despite the adaptive margin, another major difference between $L _ { w }$ and $L _ { s o f t }$ is that $L _ { s o f t }$ will use the hard negatives which are the most similar negative pairs. Although the hard negatives are helpful in improving the performance, $L _ { w }$ cannot be beneficial from it due to the existence of noisy correspondence. Specifically, it is expected that only the similarity of the true positives is larger than that of the hard negatives. However, in the case of noisy correspondence, the similarity of the false positives will also be larger than that of the hard negatives, thus leading to the unavailability of the hard negatives for $L _ { w }$ during the co-divide stage. The detail of NCR is presented in Algorithm. 1. ",
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+    {
+        "type": "text",
+        "text": "3.4 Discussions on Matching Loss ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To achieve robust cross-modal matching with the refined soft labels, we design a soft Triplet loss by recasting the labels into soft margins. Recently, there are some works have been proposed to handle the soft labels in the matching model. For example, Wray et al. [38] recast the soft similarity into binary labels by directly setting a threshold on the predicted similarity. Kim et al. [15] proposes a log-ratio matching loss with a regularization defined by the label distance ratio, which is computed by the continuous labels. Liu et al. [25] introduces the hubness problem in image-text matching and proposes to consider all samples in a mini-batch and weights them according to both local and global statistics. Wray et al. [38] recasts the soft similarity into binary labels by directly setting a threshold on the predicted similarity. Different from them, NCR proposes to recast the rectified soft labels into soft margins in the triplet loss by assigning large margins to the true positive pairs and small ones to the false positive pairs. As a result and more importantly, our loss is specifically designed to solve the noisy correspondence problem whereas existing ones are not. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 Experiment ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we carry out experiments to verify the effectiveness of NCR in robust image-text matching. In the experiments, we use three benchmark datasets including Flickr30K [42], MS-COCO [23], and Conceptual Captions [35]. Among them, Conceptual Captions is with real noisy correspondence from the wild, and Flickr30K and MS-COCO are with simulated noisy correspondence. ",
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+    {
+        "type": "text",
+        "text": "4.1 Datasets and Performance Measurements ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Three datasets are used in our experiments. To be specific, Flickr30K contains 31,000 images collected from the Flickr website with five captions each. Following [19], we use 1,000 images for validation, 1,000 images for testing, and the rest for training. MS-COCO contains 123,287 images with five captions each. We follow the data partition in [19] which consists of 113,287 training images, 5,000 validation images, and 5,000 test images. As Flickr30K and MS-COCO are well annotated, we simulate the noisy correspondence by randomly shuffling the captions of training images for a specific percentage, denoted by noise ratio. Conceptual Captions is a large-scale data consisting of 3.3M images with a single caption each. As this data set is harvested from the Internet, about $3 \\% \\sim 2 0 \\%$ correspondences are incorrect [35]. In our experiments, we use a subset of Conceptual Captions for evaluation, named CC152K. Specifically, we randomly select 150,000 samples from the training split for training, 1,000 samples from the validation split for validation, and 1,000 samples from the validation split for testing. ",
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+    {
+        "type": "text",
+        "text": "Following [19], for all images, we take the Faster-RCNN [33] detector provided by [1] to extract the top 36 region proposals of which each is encoded as a 2048-dimensional feature. For evaluation, we take the recall at K $. ( \\mathbb { R } ^ { \\ @ \\mathbb { K } ) }$ as the measurement. In short, ${ \\mathrm { R @ K } }$ is the fraction of queries for which the correct item is retrieved in the closest K points to the query. In the experiments, we report $\\mathbf { R } \\ @ 1$ , $\\mathbf { R } @ 5$ , and $\\mathbb { R } \\ @ 1 0$ for a comprehensive evaluation. ",
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+    {
+        "type": "text",
+        "text": "4.2 Implementation Details ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "NCR is a general framework which could enable almost all existing cross-modal matching methods robust against noisy correspondence. To verify the effectiveness of our framework, SGR [7] is chosen to guarantee the robustness because it is the state of the art in image-text matching. In brief, the image regions and words are projected into a shared embedding space through a full-connected layer $( i . e . , f )$ and a Bi-GRU [34] $( i . e . , g )$ , respectively. For the similarity function $S$ , it will compute the similarity between the given image and text by combining the local and global features with the help of a graph reasoning technique proposed in [18]. Due to the space limitation, we leave more details and results in the supplemental material. ",
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+    {
+        "type": "text",
+        "text": "We train our network using the Adam optimizer [16] with the default parameters and a batch size of 128. For fair comparisons, the networks $f$ and $g$ are the same with SGR, i.e., the word embedding size is 300 and the joint embedding space size is 2048. In addition, we fix the margin $\\alpha = 0 . 2$ and $m = 1 0$ for the soft margin through the experiments. At the inference stage, we average the similarities predicted by network $A$ and $B$ for the retrieval evaluation. To avoid overfitting, we choose the best checkpoint in terms of the sum of the recalls on the validation set. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 Comparisons with State of The Arts ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            787,
+            464,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we conduct comparisons on the three datasets. The baselines include SCAN [19], VSRN [21], IMRAM [5], SGRAF, SGR and SAF [7]. For Flickr30K and MS-COCO, we report the results with three different noise ratios, i.e., $0 \\%$ , $20 \\%$ , and $50 \\%$ . In addition, we also report the results of SGR on the clean Flickr30K and MS-COCO by discarding the noisy pairs, denoted by SGR-C. Clearly, SGR-C is a quite strong baseline since the used data does not contain noisy correspondence. We do not report the results of SGRAF and SAF on the clean datasets since our framework only extends SGR in this paper. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "When the noise rate is $0 \\%$ , we directly refer to the results reported in the corresponding papers. For the noisy cases, we train the baseline models with the recommended settings three times and report the best result. Note that for SGR, we found it is very sensitive to the noisy correspondence, as shown in Table. 2. To obtain a desirable result, we experimentally employ a pre-training process to SGR (denoted by ${ \\mathrm { S G R } } ^ { * }$ ), namely, training the model with the vanilla triplet loss without hard negatives and then following the standard pipeline of SGR. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d99bdf6fd685506496cedfd712a6e748c8990051ed511e94b16895f8834230e.jpg",
+        "table_caption": [
+            "Table 1: Image-Text Retrieval on Flickr30K and MS-COCO 1K. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td colspan=\"5\">Flickr30K</td><td colspan=\"6\">MS-COCO</td></tr><tr><td colspan=\"3\">Image→Text</td><td colspan=\"3\">Text→Image</td><td colspan=\"2\">Image→Text</td><td colspan=\"3\">Text→Image</td></tr><tr><td>Noise</td><td>Methods</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1 R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td rowspan=\"7\">0%</td><td>SCAN</td><td>67.4</td><td>90.3</td><td>95.8</td><td>48.6</td><td>77.7</td><td>85.2</td><td>69.2 93.6</td><td>97.6</td><td>56.0</td><td>86.5</td><td>93.5</td></tr><tr><td>VSRN</td><td>71.3</td><td>90.6</td><td>96.0</td><td>54.7</td><td>81.8 88.2</td><td>76.2</td><td>94.8</td><td>98.2</td><td>62.8</td><td>89.7</td><td>95.1</td></tr><tr><td>IMRAM</td><td>74.1</td><td>93.0</td><td>96.6</td><td>53.9</td><td>79.4</td><td>87.2</td><td>76.7 95.6</td><td>98.5</td><td>61.7</td><td>89.1</td><td>95.0</td></tr><tr><td>SAF</td><td>73.7</td><td>93.3</td><td>96.3</td><td>56.1</td><td>81.5</td><td>88.0</td><td>76.1 95.4</td><td>98.3</td><td>61.8</td><td>89.4</td><td>95.3</td></tr><tr><td>SGR</td><td>75.2</td><td>93.3</td><td>96.6</td><td>56.2</td><td>81.0</td><td>86.5</td><td>78.0 95.8</td><td>98.2</td><td>61.4</td><td>89.3</td><td>95.4</td></tr><tr><td>SGRAF NCR</td><td>77.8</td><td>94.1</td><td>97.4</td><td>58.5</td><td>83.0</td><td>88.8</td><td>79.6 96.2</td><td>98.5</td><td>63.2</td><td>90.7</td><td>96.1</td></tr><tr><td></td><td>77.3</td><td>94.0</td><td>97.5</td><td>59.6</td><td>84.4</td><td>89.9</td><td>78.7 95.8</td><td>98.5</td><td>63.3</td><td>90.4</td><td>95.8</td></tr><tr><td rowspan=\"7\">20%</td><td>SCAN</td><td>59.1</td><td>83.4</td><td>90.4</td><td>36.6</td><td>67.0</td><td>77.5 78.2</td><td>66.2 25.1</td><td>91.0 96.4</td><td>45.0</td><td>80.2</td><td>89.3</td></tr><tr><td>VSRN IMRAM</td><td>58.1 63.0</td><td>82.6</td><td>89.3</td><td>40.7</td><td>68.7</td><td></td><td>59.0</td><td>74.8</td><td>17.6</td><td>49.0</td><td>64.1</td></tr><tr><td>SAF</td><td>51.0</td><td>86.0</td><td>91.3</td><td>41.4</td><td>71.2</td><td>80.5 68.6</td><td>92.8</td><td>97.6</td><td>55.7</td><td>85.0</td><td>91.0</td></tr><tr><td>SGR*</td><td>62.8</td><td>79.3</td><td>88.0</td><td>38.3</td><td>66.5</td><td>76.2</td><td>67.3 92.5 91.7</td><td>96.6 96.2</td><td>53.4 52.9</td><td>84.5</td><td>92.4</td></tr><tr><td>SGR-C</td><td>72.8</td><td>86.2 90.8</td><td>92.2 95.4</td><td>44.4 56.4</td><td>72.3</td><td>80.4</td><td>67.8</td><td></td><td></td><td>83.5</td><td>90.1</td></tr><tr><td>NCR</td><td>75.0</td><td>93.9</td><td>97.5</td><td></td><td>82.1</td><td>88.6</td><td>75.4 95.2</td><td>97.9</td><td>60.1</td><td>88.5 89.3</td><td>94.8</td></tr><tr><td></td><td></td><td></td><td></td><td>58.3</td><td>83.0</td><td>89.0</td><td>77.7</td><td>95.5</td><td>98.2</td><td>62.5</td><td></td><td>95.3 21.0</td></tr><tr><td rowspan=\"7\">50%</td><td>SCAN</td><td>27.7</td><td>57.6</td><td>68.8</td><td>16.2</td><td>39.3</td><td>49.8</td><td>40.8</td><td>73.5 84.9</td><td>5.4</td><td></td><td>15.1</td></tr><tr><td>VSRN</td><td>14.3</td><td>37.6</td><td>50.0</td><td>12.1</td><td>30.0</td><td>39.4</td><td>23.5 54.7</td><td>69.3</td><td>16.0</td><td>47.8</td><td>65.9</td></tr><tr><td>IMRAM</td><td>9.1</td><td>26.6</td><td>38.2</td><td>2.7</td><td>8.4</td><td>12.7</td><td>21.3 60.2</td><td>75.9</td><td>22.3</td><td>52.8</td><td>64.3</td></tr><tr><td>SAF</td><td>30.3</td><td>63.6</td><td>75.4</td><td>27.9</td><td>53.7</td><td>65.1</td><td>30.4 67.8</td><td>82.3</td><td>33.5</td><td>69.0</td><td>82.8</td></tr><tr><td>SGR*</td><td>36.9</td><td>68.1</td><td>80.2</td><td>29.3</td><td>56.2</td><td>67.0</td><td>60.6 87.4</td><td>93.6</td><td>46.0</td><td>74.2</td><td>79.0</td></tr><tr><td>SGR-C</td><td>69.8</td><td>90.3</td><td>94.8</td><td>50.1</td><td>77.5</td><td>85.2</td><td>71.7 94.1</td><td>97.7</td><td>57.0</td><td>86.6</td><td>93.7</td></tr><tr><td>NCR</td><td>72.9</td><td>93.0</td><td>96.3</td><td>54.3</td><td>79.8</td><td>86.5</td><td>74.6</td><td>94.6</td><td>97.8 59.1</td><td>87.8</td><td>94.5</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on Flickr30K & MS-COCO. Table 1 shows the quantitative results on Flickr30K and MS-COCO. Note that for MS-COCO, we only report the results by averaging over 5 folds of 1K test images due to space limitation, and leave the results on the full 5K test images in the supplemental material. From the results, one could observe that NCR is competitive to SGRAF in the noisefree case, namely, NCR could achieve state-of-the-art performance even though it is proposed to achieve robustness. When the data is contaminated by the noisy correspondence, NCR remarkably outperforms all the baselines by a large margin. Even comparing with SGR-C which is trained on the clean data, NCR improves $\\mathbb { R } \\ @ 1$ by $2 . 2 \\%$ , $3 . 1 \\%$ , $2 . 3 \\%$ , and $2 . 9 \\%$ in these four valuations. ",
+        "bbox": [
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+            484,
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+            597
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on CC152K. Table 2 shows the quantitative results on the CC152K which is with real noisy correspondences. From the results, one could see that our NCR consistently outperforms the evaluated models by a considerable margin in terms of all metrics. Specifically, NCR is $4 . 5 \\%$ and $5 . 4 \\%$ higher than the best baseline in terms of $\\mathbf { R } \\ @ 1$ in text and image retrieval, respectively. Moreover, the large performance gap between SGR and $\\operatorname { S G R } ^ { * }$ shows the noise sensitivity of the original SGR. ",
+        "bbox": [
+            174,
+            602,
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+            671
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/ee4da61e843468259b8e4aa2049d759bd9f314376e9352e94ed997753b97cea1.jpg",
+        "table_caption": [
+            "Table 2: Image-Text Retrieval on CC152K. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Image→ Text</td><td colspan=\"3\">Text→Image</td></tr><tr><td>Methods</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td>SCAN (ECCV&#x27;18)</td><td>30.5</td><td>55.3</td><td>65.3</td><td>26.9</td><td>53.0</td><td>64.7</td></tr><tr><td>VSRN (ICCV&#x27;19)</td><td>32.6</td><td>61.3</td><td>70.5</td><td>32.5</td><td>59.4</td><td>70.4</td></tr><tr><td>IMRAM (CVPR&#x27;20)</td><td>33.1</td><td>57.6</td><td>68.1</td><td>29.0</td><td>56.8</td><td>67.4</td></tr><tr><td>SAF (AAAI&#x27;21)</td><td>31.7</td><td>59.3</td><td>68.2</td><td>31.9</td><td>59.0</td><td>67.9</td></tr><tr><td>SGR (AAAI&#x27;21)</td><td>11.3</td><td>29.7</td><td>39.6</td><td>13.1</td><td>30.1</td><td>41.6</td></tr><tr><td>SGR* (AAAI&#x27;21)</td><td>35.0</td><td>63.4</td><td>73.3</td><td>34.9</td><td>63.0</td><td>72.8</td></tr><tr><td>NCR</td><td>39.5</td><td>64.5</td><td>73.5</td><td>40.3</td><td>64.6</td><td>73.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 Comparison to pre-trained model ",
+        "text_level": 1,
+        "bbox": [
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+            857,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we perform comparison to the large pre-trained model CLIP [31]. In brief, CLIP is trained on a massive dataset harvested from the Internet and thus presumably has a lot of noisy image-text pairs. Such a comparison is helpful in understanding, big data based model (CLIP) or noisy correspondence modeling technique (NCR), which one is more favorable to handle the mismatching problem. More specifically, CLIP claims that using hundreds of million data could ignore the existence of possible noise, while we believe that a well-designed algorithm is essential to solve the noisy correspondence. Noticed, although some existing works including CLIP have slightly/indirectly realized the existence of the noise, NONE of them explicitly give a solution to solve this problem and explores the characteristics of the noise correspondence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            90,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In the experiments, we conduct CLIP on the MS-COCO dataset under the following two settings, i.e., Zero-shot and Fine-tune. In brief, the first setting directly uses the released pre-trained CLIP to perform inference on MS-COCO, and the second fine-tunes the pre-trained model using the noisy training data of MS-COCO. As CLIP only released some pre-trained models and inference code 3, we use the non-official code 4 to fine-tune the model with 32 epochs for the fine-tune setting. Note that CLIP (ViT-L/14†) is unreleased and we report the results from the original paper [31]. One could observe that, although CLIP utilizes 400 million image-text pairs for pre-training, its performance inevitably degenerates during fine-tuning. In contrast, NCR achieves the matching performance with the presence of noisy correspondence, indicating the necessity of algorithm design. ",
+        "bbox": [
+            173,
+            194,
+            825,
+            319
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/4917d0b351b31f4228103f7f7cdef5414b86ca797b9f66407b7c2675314e7610.jpg",
+        "table_caption": [
+            "Table 3: Comparison with CLIP on MS-COCO 5K. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"3\">Image→ Text</td><td colspan=\"3\">Text→Image</td></tr><tr><td>Noise Ratio</td><td>Methods</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td rowspan=\"2\">0%, Zero-Shot</td><td>CLIP (ViT-L/14+)</td><td>58.4</td><td>81.5</td><td>88.1</td><td>37.8</td><td>62.4</td><td>72.2</td></tr><tr><td>CLIP (ViT-B/32)</td><td>50.2</td><td>74.6</td><td>83.6</td><td>30.4</td><td>56.0</td><td>66.8</td></tr><tr><td rowspan=\"2\">20%,Fine-tune</td><td>NCR</td><td>58.2</td><td>84.2</td><td>91.5</td><td>41.7</td><td>71.0</td><td>81.3</td></tr><tr><td>CLIP (ViT-B/32) NCR</td><td>21.4 56.9</td><td>49.6 83.6</td><td>63.3 91.0</td><td>14.8 40.6</td><td>37.6 69.8</td><td>49.6 80.1</td></tr><tr><td rowspan=\"2\">50%, Fine-tune</td><td>CLIP (ViT-B/32)</td><td>10.9</td><td>27.8</td><td>38.3</td><td>7.8</td><td>19.5</td><td></td></tr><tr><td>NCR</td><td>53.1</td><td>80.7</td><td>88.5</td><td>37.9</td><td>66.6</td><td>26.8 77.8</td></tr></table>",
+        "bbox": [
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+            361,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5 Experimental Analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            535,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first conduct experiments to show the robustness and generalizability of the proposed method. Then, we investigate the effect of co-divide and co-rectify with the visualization results. After that, we carry out the ablation study to verify different components of NCR. Finally, we visually demonstrate some noisy cases detected by NCR. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5.1 Study on Robustness and Generalizability ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            631,
+            516,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To show the robustness of NCR, we conduct experiments on Flickr30K by increasing the noise ratio from $0 \\%$ to $6 0 \\%$ with an interval of $1 0 \\%$ . In addition, to verify the generalizability of NCR to other image-text matching methods, we extend SCAN [19] by NCR, denoted by NCR-SCAN. Fig. 4 shows that NCR and NCR-SCAN perform more stable than SGR and SCAN with increasing noise ratio. Moreover, NCR (NCR-SCAN) is remarkably superior to SGR (SCAN) in all tests, which shows the generalizability of NCR. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5.2 Visualization on the Co-divide and Co-rectify ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            752,
+            539,
+            768
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To further investigate the influence of the co-divide and co-rectify modules in our method, we carry out experiments on the Flickr30K dataset by visualizing the per-sample loss distribution and the model predictions on the noisy data, where the noisy ratio is $20 \\%$ . For better visualization, here we show the result of NCR-SCAN and leave the result of NCR in the supplemental material. As shown in Fig. 3(b), the loss of most noisy samples is larger than the clean loss, which verifies the memorization effect of DNNs. By fitting the per-sample loss with GMM, NCR could effectively divide the data into clean and noisy splits. Regarding the analysis on the co-rectify, Fig. 3(c) shows that the rectified soft labels of most clean pairs range into [0.3, 1] and those of most noisy pairs range into [0, 0.5]. In other words, one could enforce the similarity of true positives larger than that of the negatives during training, thus eliminating the negative impact of the noisy correspondence. ",
+        "bbox": [
+            173,
+            776,
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+            875
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/0503d947414e2b32158fb766cbf1a3b87cabce246d93f4abb1f73ad11d4aa0c3.jpg",
+        "image_caption": [
+            "Figure 3: (a) Retrieval performance of NCR and NCR-SCAN on Flickr30K with varying noise ratio. (b) Per-sample loss distribution and GMM fitting visualization after warmup. (c) Model predictions on the noisy subset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            79,
+            812,
+            213
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            284,
+            825,
+            327
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.5.3 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            344,
+            331,
+            359
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this section, we carry out the ablation study on the Flickr30K with the noise ratio of $50 \\%$ . As shown in Table 4, all these three components are important to achieving encouraging results. ",
+        "bbox": [
+            174,
+            368,
+            823,
+            398
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/fe7374156c5916db0ea3f6530f15940fb5c71d44d889dd8d48eeb9982fd8b3df.jpg",
+        "table_caption": [
+            "Table 4: Ablation studies on Flickr30K with $50 \\%$ noise. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Method</td><td colspan=\"3\">Image →Text</td><td colspan=\"3\">Text→Image</td></tr><tr><td>Co-divide</td><td>Co-rectify</td><td>Warmup</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td>?</td><td>√</td><td>&lt;&lt;&gt;</td><td>72.9</td><td>93.0</td><td>96.3</td><td>54.3</td><td>79.8</td><td>86.5</td></tr><tr><td></td><td></td><td></td><td>71.4</td><td>90.8</td><td>95.7</td><td>54.1</td><td>80.3</td><td>86.5</td></tr><tr><td></td><td>?</td><td></td><td>16.0</td><td>38.4</td><td>51.7</td><td>12.6</td><td>31.4</td><td>42.8</td></tr><tr><td>√</td><td></td><td></td><td>0.3</td><td>0.6</td><td>1.0</td><td>0.2</td><td>0.5</td><td>1.1</td></tr></table>",
+        "bbox": [
+            205,
+            425,
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+            536
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.5.4 Noisy Samples ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            558,
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+            573
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fig. 4 shows some noisy CC152K examples identified by NCR. As shown, the first four image-text pairs are completely unrelated, which will be successfully detected by NCR. For the last pair, it will also be detected as noisy correspondence even though the visual and textual modalities are correlated at a coarse-grained level, e.g., both the image and text involve “beach\". ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/c2ea469f89d768984f1c578d0e0317f6d920992f64f93e122eb3703e75d9c2da.jpg",
+        "image_caption": [
+            "Figure 4: Some noisy examples correctly divided by NCR. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper could be the first attempt to study a new problem in cross-modal matching, i.e., the noisy correspondence which could be a potential new direction in noise label. To solve this problem in cross-modal matching, we propose rectifying the noisy correspondence by an adaptive prediction function and a novel triplet loss with a soft margin to achieve robust cross-modal matching. Extensive experiments verify the effectiveness of the proposed method in handling synthesized and real noisy correspondences. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Broader Impact Statement ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            398,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Cross-modal matching is a fundamental topic in multimodal learning, which could be applied to a wide range of applications including data retrieval, recommender systems, and vision-and-language understanding. This work could be one of the first works to aware of the importance and existence of the noisy correspondence problem in numerous applications. There are many benefits to solving the noisy correspondence problem, e.g., reducing the costs for manually annotating and aligning data; more data could be collected and used even though some of them are incorrectly aligned. Besides the benefits, it should pay attention on the potential negative impacts including but not limited to 1) The risk of automation bias [28] for decision making from the data bias, especially in aviation, health care, and autonomous vehicles. 2) The job loss caused by the NCR since it makes possibility to automatically correct the noisy correspondence, thus remarkably reducing the cost of human labor. We would encourage further work to understand and mitigate the above biases and risks. ",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            291,
+            338,
+            309
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The authors would thank to the anonymous reviewers whose valuable suggestions and constructive comments remarkably improve this work. This work was supported in part by the National Key R&D Program of China under Grant 2020AAA0104500; in part by the Key Research and Development Program of Sichuan Province under Grant 2019YFG0497; in part by NFSC under Grant U21B2040, 62176171, U19A2078, 61625204, and 61836006; in part by the Fund of Sichuan University Tomorrow Advancing Life. ",
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            425,
+            266,
+            441
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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Pattern Recognition, 39(4):695–706, 2006.   \n[31] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. arXiv preprint arXiv:2103.00020, 2021.   \n[32] Scott Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich. Training deep neural networks on noisy labels with bootstrapping. arXiv preprint arXiv:1412.6596, 2014.   \n[33] Shaoqing Ren, Kaiming He, Ross B Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In NIPS, 2015.   \n[34] Mike Schuster and Kuldip K Paliwal. Bidirectional recurrent neural networks. IEEE transactions on Signal Processing, 45(11):2673–2681, 1997.   \n[35] Piyush Sharma, Nan Ding, Sebastian Goodman, and Radu Soricut. 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parse/train/SypU81Ole/SypU81Ole.md

@@ -0,0 +1,189 @@
+# SAMPLING GENERATIVE NETWORKS
+
+Tom White   
+School of Design   
+Victoria University of Wellington   
+Wellington, New Zealand   
+tom.white@vuw.ac.nz
+
+# ABSTRACT
+
+We introduce several techniques for sampling and visualizing the latent spaces of generative models. Replacing linear interpolation with spherical linear interpolation prevents diverging from a model’s prior distribution and produces sharper samples. J-Diagrams and MINE grids are introduced as visualizations of manifolds created by analogies and nearest neighbors. We demonstrate two new techniques for deriving attribute vectors: bias-corrected vectors with data replication and synthetic vectors with data augmentation. Binary classification using attribute vectors is presented as a technique supporting quantitative analysis of the latent space. Most techniques are intended to be independent of model type and examples are shown on both Variational Autoencoders and Generative Adversarial Networks.
+
+# 1 INTRODUCTION
+
+Generative models are a popular approach to unsupervised machine learning. Generative neural network models are trained to produce data samples that resemble the training set. Because the number of model parameters is significantly smaller than the training data, the models are forced to discover efficient data representations. These models are sampled from a set of latent variables in a high dimensional space, here called a latent space. Latent space can be sampled to generate observable data values. Learned latent representations often also allow semantic operations with vector space arithmetic (Figure 1).
+
+![](images/2555a5b238a3ab69e2c09451d964cdc3cd6249e1558a4a8be442906ffa2cf867.jpg)  
+Figure 1: Schematic of the latent space of a generative model. In the general case, a generative model includes an encoder to map from the feature space (here images of faces) into a high dimensional latent space. Vector space arithmetic can be used in the latent space to perform semantic operations. The model also includes a decoder to map from the latent space back into the feature space, where the semantic operations can be observed. If the latent space transformation is the identity function we refer to the encoding and decoding as a reconstruction of the input through the model.
+
+Generative models are often applied to datasets of images. Two popular generative models for image data are the Variational Autoencoder (VAE, Kingma & Welling, 2014) and the Generative Adversarial Network (GAN, Goodfellow et al., 2014). VAEs use the framework of probabilistic graphical models with an objective of maximizing a lower bound on the likelihood of the data. GANs instead formalize the training process as a competition between a generative network and a separate discriminative network. Though these two frameworks are very different, both construct high dimensional latent spaces that can be sampled to generate images resembling training set data. Moreover, these latent spaces are generally highly structured and can enable complex operations on the generated images by simple vector space arithmetic in the latent space (Larsen et al., 2016).
+
+Generative models are beginning to find their way out of academia and into creative applications. In this paper we present techniques for improving the visual quality of generative models that are generally independent of the model itself. These include spherical linear interpolation, visualizing analogies with J-diagrams, and generating local manifolds with MINE grids. These techniques can be combined generate low dimensional embeddings of images close to the trained manifold. These can be used for visualization and creating realistic interpolations across latent space. Also by standardizing these operations independent of model type, the latent space of different generative models are more directly comparable with each other, exposing the strengths and weaknesses of various approaches.
+
+Additionally, two new techniques for building latent space attribute vectors are introduced. On labeled datasets with correlated labels, data replication can be used to create bias-corrected vectors. Synthetic attributes vectors also can be derived via data augmentation on unlabeled data. Quantitative analysis of attribute vectors can be performed by using them as the basis for attribute binary classifiers.
+
+# 2 SAMPLING TECHNIQUES
+
+Generative models are often evaluated by examining samples from the latent space. Techniques frequently used are random sampling and linear interpolation. But often these can result in sampling the latent space from locations very far outside the manifold of probable locations.
+
+Our work has followed two useful principles when sampling the latent space of a generative model. The first is to avoid sampling from locations that are highly unlikely given the prior of the model. This technique is very well established - including being used in the original VAE paper which adjusted sampling through the inverse CDF of the Gaussian to accommodate the Gaussian prior (Kingma & Welling, 2014). The second principle is to recognize that the dimensionality of the latent space is often artificially high and may contains dead zones that are not on the manifold learned during training. This has been demonstrated for VAE models (Makhzani et al., 2016) and implies that simply matching the model’s prior will not always be sufficient to yield samples that appear to have been drawn from the training set.
+
+# 2.1 INTERPOLATION
+
+Interpolation is used to traverse between two known locations in latent space. Research on generative models often uses interpolation as a way of demonstrating that a generative model has not simply memorized the training examples (eg: Radford et al., 2015, $\ S 6 . 1 \AA ,$ ). In creative applications interpolations can be used to provide smooth transitions between two decoded images.
+
+Frequently linear interpolation is used, which is easily understood and implemented. But this is often inappropriate as the latent spaces of most generative models are high dimensional $( > 5 0$ dimensions) with a Gaussian or uniform prior. In such a space, linear interpolation traverses locations that are extremely unlikely given the prior. As a concrete example, consider a 100 dimensional space with the Gaussian prior $\scriptstyle \mu = 0$ , $\sigma { = } 1$ . Here all random vectors will generally a length very close to 10 (standard deviation $< 1$ ). However, linearly interpolating between any two will usually result in a "tent-pole" effect as the magnitude of the vector decreases from roughly 10 to 7 at the midpoint, which is over 4 standard deviations away from the expected length.
+
+Our proposed solution is to use spherical linear interpolation slerp instead of linear interpolation. We use a formula introduced by (Shoemake 85) in the context of great arc inbetweening for rotation animations:
+
+$$
+S l e r p ( q _ { 1 } , q _ { 2 } ; \mu ) = \frac { \sin { ( 1 - \mu ) \theta } } { \sin { \theta } } q _ { 1 } + \frac { \sin { \mu \theta } } { \sin { \theta } } q _ { 2 }
+$$
+
+This treats the interpolation as a great circle path on an n-dimensional hypersphere (with elevation changes). This technique has shown promising results on both VAE and GAN generative models and with both uniform and Gaussian priors (Figure 2).
+
+![](images/8aae2d9c8a78ba60fe122cdffe2e1c90eb9cc17ebb47405d844f2f462451a069.jpg)  
+Figure 2: DCGAN (Radford 15) interpolation pairs with identical endpoints and uniform prior. In each pair, the top series is linear interpolation and the bottom is spherical. Note the weak generations produced by linear interpolation at the center, which are not present in spherical interpolation.
+
+# 2.2 ANALOGY
+
+Analogy has been shown to capture regularities in continuous space models. In the latent space of some linguistic models “King – Man $^ +$ Woman” results in a vector very close to “Queen” (Mikolov et al., 2013). This technique has also been used in the context of deep generative models to solve visual analogies (Reed et al., 2015).
+
+Analogies are usually written in the form:
+
+$$
+A : B : : C : ?
+$$
+
+Such a formation answers the question “What is the result of applying the transformation A:B to C?” In a vector space the solution generally proposed is to solve the analogy using vector math:
+
+$$
+( B - A ) = ( ? - C )
+$$
+
+$$
+? = C + B - A
+$$
+
+Note that an interesting property of this solution is an implied symmetry:
+
+$$
+A : C : : B : ?
+$$
+
+Because the same terms can be rearranged:
+
+$$
+( C - A ) = ( ? - B )
+$$
+
+$$
+? = B + C - A
+$$
+
+Generative models that include an encoder for computing latent vectors given new samples allow for visual analogies. We have devised a visual representation for depicting analogies of visual generative networks called a “J-Diagram”. The J- Diagram uses interpolation across two dimensions two expose the manifold of the analogy. It also makes the symmetric nature of the analogy clear (Figure 3).
+
+The J-Diagram also serves as a reference visualization across different model settings because it is deterministically generated from images which can be held constant. This makes it a useful tool for comparing results across epochs during training, after adjusting hyperparameters, or even across completely different model types (Figure 4).
+
+![](images/e1290feaf1c831efd10ce46a564ea1977174665ca522bfad469160b29a6b92b6.jpg)  
+Figure 3: J-Diagram. The three corner images are inputs to the system, with the top left being the “source” (A) and the other two being “analogy targets” (B and C). Adjacent to each is the reconstruction resulting from running the image through both the encoder and decoder of the model. The bottom right image shows the result of applying the analogy operation $( \mathbf { B } + \mathbf { C } ) - \mathbf { A }$ . All other images are interpolations using the slerp operator. (model: VAE from Lamb 16 on CelebA)
+
+![](images/e2c12f518e7a60e7756fcff5a47ee67e605eac0dece4b8a72e38308c67903fc1.jpg)  
+Figure 4: Same J-Diagram repeated with different model type. To facilitate comparisons (and demonstrate results are not cherry-picked) inputs selected are the first 3 images of the validation set. (model: GAN from Dumoulin 16 on CelebA)
+
+# 2.3 MANIFOLD TRAVERSAL
+
+Generative models can produce a latent space that is not tightly packed, and the dimensionality of the latent space is often set artificially high. As a result, the manifold of trained examples is can be a subset of the latent space after training, resulting in dead zones in the expected prior.
+
+If the model includes an encoder, one simple way to stay on the manifold is by only using out of sample encodings (ie: any data not used in training) in the latent space. This is a useful diagnostic, but is overly restrictive in a creative application context since it prevents the model from suggesting new and novel samples from the model. However, we can recover this ability by also including the results of operations on these encoded samples that stay close to the manifold, such as interpolation, extrapolation, and analogy generation.
+
+Ideally, there would be a mechanism to discover this manifold within the latent space. In generative models with an encoder and ample out-of-sample data, we can instead precompute locations on the manifold with sufficient density, and later query for nearby points in the latent space from this known set. This offers a navigation mechanism based on hopping to nearest neighbors across a large database of encoded samples. When combined with interpolation, we call this visualization a Manifold Interpolated Neighbor Embedding (MINE). A MINE grid is useful to visualize local patches of the latent space (Figure 5).
+
+![](images/5c968d2e79efc68093073d2e767c0ac6d0414407c99248db2b212c59fc275721.jpg)  
+(b) Reconstructions are spread with interpolation to expose areas between them.   
+Figure 5: Example of local VAE manifold built using the $3 0 \mathrm { k }$ CelebA validation and test images as a dataset of out of sample features. The resulting MINE grid represents a small contiguous manifold of the larger latent space. (model: VAE from Lamb 16 on CelebA)
+
+# 3 ATTRIBUTE VECTORS
+
+Many generative models result in a latent space that is highly structured, even on purely unsupervised datasets (Radford et al., 2015). When combined with labeled data, attribute vectors can be computed using simple arithmetic. For example, a vector can be computed which represents the smile attribute, which by shorthand we call a smile vector. Following (Larsen et al., 2016), the smile vector can be computed by simply subtracting the mean vector for images without the smile attribute from the mean vector for images with the smile attribute. This smile vector can then be applied to in a positive or negative direction to manipulate this visual attribute on samples taken from latent space (Figure 6).
+
+![](images/369499ddd963f3fcfa56e86da65b1e2f780c624c8dbdfb4bec9bf195d3b0fc9f.jpg)  
+Figure 6: Traversals along the smile vector. (model: GAN from Dumoulin 16 on CelebA)
+
+# 3.1 CORRELATED LABELS
+
+The approach of building attribute vectors from means of labeled data has been noted to suffer from correlated labels (Larsen et al., 2016). While many correlations would be expected from ground truths (eg: heavy makeup and wearing lipstick) we discovered others that appear to be from sampling bias. For example, male and smiling attributes have unexpected negative correlations because women in the CelebA dataset are much more likely to be smiling than men. (Table 1).
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Male</td><td rowspan=1 colspan=1>Not Male</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>Smiling</td><td rowspan=1 colspan=1>17%</td><td rowspan=1 colspan=1>31%</td><td rowspan=1 colspan=1>48%</td></tr><tr><td rowspan=1 colspan=1>Not Smiling</td><td rowspan=1 colspan=1>25%</td><td rowspan=1 colspan=1>27%</td><td rowspan=1 colspan=1>52%</td></tr><tr><td rowspan=1 colspan=1>Total</td><td rowspan=1 colspan=1>42%</td><td rowspan=1 colspan=1>58%</td><td rowspan=1 colspan=1></td></tr></table>
+
+Table 1: Breakdown of CelebA smile versus male attributes. In the total population the smile attribute is almost balanced $4 8 \%$ smile). But separating the data further shows that those with the male attribute smile only $42 \%$ of the time while those without it smile $58 \%$ of the time.
+
+![](images/12e22a423a2190d78b89c4905870ca763f27e7d34d28d2eb69364768552065dd.jpg)  
+Figure 7: Initial attempts to build a smile vector suffered from sampling bias. The effect was that removing smiles from reconstructions (left) also added male attributes (center). By using replication to balance the data across both attributes before computing the attribute vectors, the gender bias was removed (right). (model: VAE from Lamb 16 on CelebA)
+
+In an online service we setup to automatically add and remove smiles from images1, we discovered this gender bias was visually evident in the results. Our solution was to use replication on the training data such that the dataset was balanced across attributes. This was effective because ultimately the vectors are simply summed together when computing the attribute vector (Figure 7).
+
+This balancing technique can also be applied to attributes correlated due to ground truths. Decoupling attributes allows individual effects to be applied separately. As an example, the two attributes smiling and mouth open are highly correlated in the CelebA training set (Table 2). This is not surprising, as physically most people photographed smiling would also have their mouth open. However by forcing these attributes to be balanced, we can construct two decoupled attribute vectors. This allows for more flexibility in applying each attribute separately to varying degrees (Figure 8).
+
+Table 2: CelebA smile versus open mouth attributes shows a strong symmetric correlation (greater than 3 to 1).   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Open Mouth</td><td rowspan=1 colspan=1>Not Open Mouth</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>Smiling</td><td rowspan=1 colspan=1>36%</td><td rowspan=1 colspan=1>12%</td><td rowspan=1 colspan=1>48%</td></tr><tr><td rowspan=1 colspan=1>Not Smiling</td><td rowspan=1 colspan=1>12%</td><td rowspan=1 colspan=1>40%</td><td rowspan=1 colspan=1>52%</td></tr><tr><td rowspan=1 colspan=1>Total</td><td rowspan=1 colspan=1>48%</td><td rowspan=1 colspan=1>52%</td><td rowspan=1 colspan=1></td></tr></table>
+
+![](images/69eab4601a0a204cd93f91d2d32563736b22537e7fffd0a22b831ef129643299.jpg)  
+Figure 8: Decoupling attribute vectors for smiling (x-axis) and mouth open (y-axis) allows for more flexible latent space transformations. Input shown at left with reconstruction adjacent. (model: VAE from Lamb 16 on CelebA)
+
+# 3.2 SYNTHETIC ATTRIBUTES
+
+It has been noted that samples drawn from VAE based models is that they tend to be blurry (Goodfellow et al., 2014; Larsen et al., 2015). A possible solution to this would be to discover an attribute vector for “unblur”, and then apply this as a constant offset to latent vectors before decoding. CelebA includes a blur label for each image, and so a blur attribute vector was computed and then extrapolated in the negative direction. This was found to noticeably reduce blur, but also resulted in a number of unwanted artifacts such as increased image brightness.
+
+We concluded this to be the result of human bias in labeling. Labelers appear more likely to label darker images as blurry, so this unblur vector was found to suffer from attribute correlation that also “lightened” the reconstruction. This bias could not be easily corrected because CelebA does not include a brightness label for rebalancing the data.
+
+For the blurring attribute, an algorithmic solution is available. We take a large set of images from the training set and process them through a Gaussian blur filter (figure 9). Then we run both the original image set and the blurred image set through the encoder and subtract the means of each set to compute a new attribute vector for blur. We call this a synthetic attribute vector because this label is derived from algorithmic data augmentation of the training set. This technique removes the labeler bias, is straightforward to implement, and resulted in samples closely resembling the reconstructions with less noticeable blur (figure 10).
+
+![](images/7bbf40f34aa3802b69e1e6f8f450dabf7543c69c4a76b250dbe87d112adffd28.jpg)  
+Figure 9: Zoomed detail from training images (top) and computed blurred images (bottom) were both encoded in order to determine a non-biased blur attribute vector.
+
+![](images/4c1c224c51d82c101fea05e04fb4347d0849d65fb20d5665fa371f8f5ff6f53b.jpg)  
+Figure 10: Zoomed detail of images from the validation set (top) are reconstructed (second row). Applying an offset in latent space based on the CelebA blur attribute (third row) does reduce noticeable blur from the reconstructions, but introduces visual artifacts including brightening due to attribute correlation. Applying an attribute vector instead computed from synthetic blur (bottom) yields images noticeably deblurred from the reconstructions and without unrelated artifacts.
+
+# 3.3 QUANTITATIVE ANALYSIS WITH CLASSICICATION
+
+The results of applying attribute vectors visually to images are often quite striking. But it would be useful to be able to evaluate the relative efficacy of various attribute vectors to each other or across different models and hyperparamaters. Attribute vectors have potential to be widely applicable outside of images in domain specific latent spaces, and this provides additional challenges for quantifying their performance. For example, recent work has shown the ability to use latent space model as a continuous representation of molecules (Gómez 16); though it is straightforward to generate vectors for attributes such as solubility in these spaces, the evaluation of these operations to existing molecules would appear to require specific domain knowledge.
+
+We have found that very effective classifiers can be built in latent space using the dot product of an encoded vector with a computed attribute vector (figure 11A, 11B). We have named this technique for building classifiers on latent spaces AtDot, and models trained on unsupervised data have been shown to produce strong results on CelebA across most attributes (table 3). Importantly, these binary classifiers provide a quantitative basis to evaluate attribute vectors on a related surrogate task - their ability to be the basis for a simple linear classifier. This task is one of the most established paradigms in machine learning and provides a set of established tools such as ROC curves for offering new insights into the behavior of the attribute vectors across different hyperparameters or different models (figure 11C).
+
+Table 3: Average Classification Accuracy on the CelebA dataset. Models for both VAE AtDot and GAN AtDot use unsupervised training with attribute vectors computed on the training set data after model training was completed. VAE AtDot from Lamb 16 with discriminative regularization disabled and GAN AtVec from Dumoulin 16. Other results are [1] Kumar 08, [2] Zhang 14, [3] Li 13, and [4] LeCun 94 as reported in Ehrlich 16.   
+
+<table><tr><td>Approach /Attribute</td><td>[1]</td><td>[2] W [2] L</td><td>[3] ANet</td><td>[4] LNet</td><td>[4] Full</td><td>VAE AtDot</td><td>GAN AtDot</td></tr><tr><td>5 0.C. Shadow</td><td>85</td><td>82 88</td><td>86</td><td>88</td><td>91</td><td>88</td><td>89</td></tr><tr><td>Arched Eyebrow</td><td>76</td><td>73 78</td><td>75</td><td>74</td><td>79</td><td>75</td><td>75</td></tr><tr><td>Attractive</td><td>78</td><td>77 81</td><td>79</td><td>77</td><td>81</td><td>71</td><td>69</td></tr><tr><td>Bags Under Eye</td><td>76</td><td>71 79</td><td>77</td><td>73</td><td>79</td><td>80</td><td>79</td></tr><tr><td>Bald</td><td>89</td><td>92 96</td><td>92</td><td>95</td><td>98</td><td>98</td><td>98</td></tr><tr><td>Bangs</td><td>88</td><td>89 92</td><td>94</td><td>92</td><td>95</td><td>90</td><td>90</td></tr><tr><td>Big Lips</td><td>64</td><td>61 67</td><td>63</td><td>66</td><td>68</td><td>85</td><td>76</td></tr><tr><td>Big Nose</td><td>74</td><td>70 75</td><td>74</td><td>75</td><td>78</td><td>77</td><td>77</td></tr><tr><td>Black Hair</td><td>70</td><td>74 85</td><td>77</td><td>84</td><td>88</td><td>83</td><td>81</td></tr><tr><td>Blond Hair</td><td>80</td><td>81 93</td><td>86</td><td>91</td><td>95</td><td>87</td><td>90</td></tr><tr><td>Blurry</td><td>81</td><td>77 86</td><td>83</td><td>80</td><td>84</td><td>95</td><td>95</td></tr><tr><td>Brown Hair</td><td>60</td><td>69 77</td><td>74</td><td>78</td><td>80</td><td>76</td><td>81</td></tr><tr><td>BushyEyebrow</td><td>80</td><td>76 86</td><td>80</td><td>85</td><td>90</td><td>87</td><td>86</td></tr><tr><td>Chubby</td><td>86</td><td>82 86</td><td>86</td><td>86</td><td>91</td><td>94</td><td>94</td></tr><tr><td>Double Chin</td><td>88</td><td>85 88</td><td>90</td><td>88</td><td>92</td><td>95</td><td>95</td></tr><tr><td>Eyeglasses</td><td>98</td><td>94 98</td><td>96</td><td>96</td><td>99</td><td>95</td><td>94</td></tr><tr><td>Goatee</td><td>93</td><td>86 93</td><td>92</td><td>92</td><td>95</td><td>93</td><td>94</td></tr><tr><td>Gray Hair</td><td>90</td><td>88 94</td><td>93</td><td>93</td><td>97</td><td>95</td><td>96</td></tr><tr><td>Heavy Makeup</td><td>85</td><td>84 90</td><td>87</td><td>85</td><td>90</td><td>76</td><td>75</td></tr><tr><td>High Cheekbone</td><td>84</td><td>80 86</td><td>85</td><td>84</td><td>87</td><td>81</td><td>68</td></tr><tr><td>Male</td><td>91</td><td>93 97</td><td>95</td><td>94</td><td>98</td><td>81</td><td>80</td></tr><tr><td>Mouth Open</td><td>87</td><td>82 93</td><td>85</td><td>86</td><td>92</td><td>78</td><td>67</td></tr><tr><td>Mustache</td><td>91</td><td>83 93</td><td>87</td><td>91</td><td>95</td><td>95</td><td>96</td></tr><tr><td>Narrow Eyes</td><td>82</td><td>79 84</td><td>83</td><td>77</td><td>81</td><td>93</td><td>89</td></tr><tr><td>No Beard</td><td>90</td><td>87 93</td><td>91</td><td>92</td><td>95</td><td>85</td><td>84</td></tr><tr><td>Oval Face</td><td>64</td><td>62 65</td><td>65</td><td>63</td><td>66</td><td>72</td><td>71</td></tr><tr><td>Pale Skin</td><td>83</td><td>84 91</td><td>89</td><td>87</td><td>91</td><td>96</td><td>96</td></tr><tr><td>Pointy Nose</td><td>68</td><td>65</td><td>71 67</td><td>70</td><td>72</td><td>72</td><td>72</td></tr><tr><td>Recede Hair</td><td>76</td><td>82</td><td>85 84</td><td>85</td><td>89</td><td>93</td><td>92</td></tr><tr><td>Rosy Cheeks</td><td>84</td><td>81</td><td>87 85</td><td>87</td><td>90</td><td>93</td><td>93</td></tr><tr><td>Sideburns</td><td>94</td><td>90</td><td>93 94</td><td>91</td><td>96</td><td>94</td><td>94</td></tr><tr><td>Smiling</td><td>89</td><td>89</td><td>92 92</td><td>88</td><td>92</td><td>87</td><td>68</td></tr><tr><td>Straight Hair</td><td>63</td><td>67</td><td>69 70</td><td>69</td><td>73</td><td>80</td><td>79</td></tr><tr><td>Wavy Hair</td><td>73</td><td>76</td><td>77 79</td><td>75</td><td>80</td><td>75</td><td>75</td></tr><tr><td>Earring</td><td>73</td><td>72</td><td>78 77</td><td>78</td><td>82</td><td>81</td><td>81</td></tr><tr><td>Hat</td><td>89</td><td>91</td><td>96 93</td><td>96</td><td>99</td><td>96</td><td>97</td></tr><tr><td>Lipstick</td><td>89</td><td>88</td><td>93 91</td><td>90</td><td>93</td><td>79</td><td>77</td></tr><tr><td>Necklace</td><td>68</td><td>67</td><td>67 70</td><td>68</td><td>71</td><td>88</td><td>88</td></tr><tr><td>Necktie</td><td>86</td><td>88</td><td>91 90</td><td>86</td><td>93</td><td>93</td><td>92</td></tr><tr><td>Young</td><td>80</td><td>77</td><td>84 81</td><td>83</td><td>87</td><td>78</td><td>81</td></tr><tr><td>Average</td><td>81</td><td>79 85</td><td>83</td><td>83</td><td>87</td><td>86</td><td>84</td></tr></table>
+
+![](images/23124b836cb1a63feda2684bd36b5d1d066add8214b0fa371e200f2960eb088d.jpg)  
+Figure 11: Example of using an attribute vector for classification. A smile vector is first constructed in latent space from labeled training data. By taking the dot product of this smile vector with the encoded representation of any face, a scalar smile score is generated (a). This score can be the basis for a binary classifier. The histogram shows the distribution of positive (green) and negative (red) scores on the CelebA validation set (b). The ROC curve for a smile vector based binary classifier is also shown (c).
+
+# 4 FUTURE WORK
+
+Software to support most techniques presented in this paper is included in a python software library that can be used with various generative models2. We hope to continue to improve the library so that the techniques are applicable across a broad range of generative models. Attribute vector based classifiers also offer a promising way to evaluate the suitability of attribute vectors in domains outside of images.
+
+We are investigating constructing a specially constructed prior on the latent space such that interpolations could be linear. This would simplify many of the latent space operations and might enable new types of operations.
+
+Given sufficient test data, the extent to which an encoded dataset deviates from the expected prior should be quantifiable. Developing such a metric would be useful in understanding the structure of the different latent spaces including probability that random samples fall outside of the expected manifold of encoded data.
+
+# ACKNOWLEDGMENTS
+
+I am thankful for the constructive feedback from readers including Ehud Ben-Reuven, Zachary Lipton, and Alex Champandard. I thank Victoria University of Wellington School of Design for supporting research on Creative Intelligence. I also thank the vibrant machine learning and creative coding communities on twitter for their support and encouragement.
+
+# REFERENCES
+
+Dumoulin, Vincent, Belghazi, Ishmael, Poole, Ben, Lamb, Alex, Arjovsky, Martin, Mastropietro, Olivier Courville, Aaron. Adversarially Learned Inference. https://arxiv.org/abs/1606.00704 2016 Ehrlich, Max, Shields, Timothy J., Almaev, Timur, Amer, Mohamed R. Facial Attributes Classification Using Multi-Task Representation Learning. The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops. 2016.   
+Goodfellow, Ian, Pouget-Abadie, Jean, Mirza, Mehdi, Xu, Bing, Warde- Farley, David, Ozair, Sherjil, Courville, Aaron, and Bengio, Yoshua. Generative adversarial nets. In Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., and Weinberger, K.Q. (eds.), Advances in Neural Information Processing Systems 27, pp. 2672–2680. Curran Associates, Inc., 2014.   
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